Тест Ферма для 341 по основанию 2 и для 561 по основаниям 2, 5 и 7. Где критерий объявляет число простым и чем число Кармайкла отличается от обычного псевдопростого?

Малая теорема Ферма даёт лишь необходимое условие. Для 341 = 11 · 31 след возведения 2 в степень 340 по модулю 341 сходится к единице, и тест пройден: перед нами псевдопростое по основанию 2. Для 561 = 3 · 11 · 17 тест проходит по основаниям 2, 5 и 7 — это число Кармайкла, обманывающее каждое взаимно простое основание. Не найдя делителя, модель заключает «простое» обоими случаями, и это ровно граница критерия, а не ошибка вычисления: каждое звено удвоения в следе движок пересчитывает и проверяет сам.

This is an assistant explanation, not a calculation result. Check the grounds and sources below.

At a glance

Select a result to explore its grounds

Простота 7 по тому же критериюConditionEstablishedExecution: ComputedLogical result: Established 341 проходит тест по основанию 2ConditionEstablishedExecution: ComputedLogical result: Established 341 объявлено простым без делителяConditionEstablishedExecution: ComputedLogical result: Established 561 по основанию 2ConditionEstablishedExecution: ComputedLogical result: Established 561 по основанию 5ConditionEstablishedExecution: ComputedLogical result: Established 561 по основанию 7ConditionEstablishedExecution: ComputedLogical result: Established 561 объявлено простымConditionEstablishedExecution: ComputedLogical result: Established

Detailed analysis

Condition

Простота 7 по тому же критерию

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

numerus primus

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 7basis: 2
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 6
residuum minimum potestatis exponentis dati
exponens valor
1 2
2 4
3 1
6 1

exponens	valor
1	2
2	4
3	1
6	1

Package: Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

Additional details

Include proof: Yes

Original data · JSON

{
  "args": [
    7
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        7,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        6
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        3,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        6,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "primus",
  "proof": true
}

Why this resultApplied rules and conditions

Derivation path12 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 7; basis: 2
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 2
case fact
3
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 4
case fact
5
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 3; valor: 1
case fact
7
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 3
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
8
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 6; valor: 1
case fact
9
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 6
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
10
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
11
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
numerus primus: modulus: 7
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1
rule
12
Query evaluation
query

verified by the engine: 7 · case fact: 5 · Full graph: 16 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation1

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus

Derived result for this query

numerus primus
modulus: 7

Other derived facts6

probatio
pr: probatio
Subject shared by the facts below
casus propositus
catena potestatum ad exponentem usque probata
exponens
1
2
3
6
probatio Fermatiana transacta: potestas p-1 unitati congrua

exponens
1
2
3
6

casus propositus
pr
urn:case:flt:probatio

catena potestatum ad exponentem usque probata
pr	exponens
urn:case:flt:probatio	1
urn:case:flt:probatio	2
urn:case:flt:probatio	3
urn:case:flt:probatio	6

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr
urn:case:flt:probatio

numerus primus
modulus
7

0 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet7Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 7 layer

Proof nodes: 16 · assertion 5, rule_application 8, candidate_closure 1, constraint_check 1, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1

attributes

assertion: fact-1

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
7
Type
value
type
name
Integer
value
2

Type

literal

Polarity

positive

Condition

propositum

evidence

—

Identifier

fact-1

Type

assertion

Premises

—

sourceAnchors

—

rule_application · urn:proof:apply:CasusPropositus:f12a3f0ef43d198f93d07f0ce6e6d997729b416e0ae1f6b37ab109cca800cd4f

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref

Type

literal

Polarity

positive

Condition

casus_propositus

evidence

—

Identifier

f12a3f0ef43d198f93d07f0ce6e6d997729b416e0ae1f6b37ab109cca800cd4f

Type

rule_application

Premises

fact-1

Rule

CasusPropositus

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

assertion · urn:proof:assert:urn:mcp:case#fact-3

attributes

assertion: fact-3

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
1
Type
value
type
name
Integer
value
2

Type

literal

Polarity

positive

Condition

residuum

evidence

—

Identifier

fact-3

Type

assertion

Premises

—

sourceAnchors

—

rule_application · urn:proof:apply:CatenaInitium:db438b0b14fb7a6e300fd430228baeaecc860d5218c2230d48b225aa90f2ca02

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
1

Type

literal

Polarity

positive

Condition

catena

evidence

—

Identifier

db438b0b14fb7a6e300fd430228baeaecc860d5218c2230d48b225aa90f2ca02

Type

rule_application

Premises

fact-1
fact-3

Rule

CatenaInitium

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

assertion · urn:proof:assert:urn:mcp:case#fact-4

attributes

assertion: fact-4

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
2
Type
value
type
name
Integer
value
4

Type

literal

Polarity

positive

Condition

residuum

evidence

—

Identifier

fact-4

Type

assertion

Premises

—

sourceAnchors

—

rule_application · urn:proof:apply:CatenaDuplicata:b9cec46229dfe374c4a6a89507e81270c9d73a63a4f430173853872396334111

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
2

Type

literal

Polarity

positive

Condition

catena

evidence

—

Identifier

b9cec46229dfe374c4a6a89507e81270c9d73a63a4f430173853872396334111

Type

rule_application

Premises

db438b0b14fb7a6e300fd430228baeaecc860d5218c2230d48b225aa90f2ca02
fact-1
fact-3
fact-4

Rule

CatenaDuplicata

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

rule_application · urn:proof:apply:CatenaGradus:46c433109e945de4f2ff86c9f8b0b44d5251ba466d2758cd17c8a22a88cbdfdc

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
2

Type

literal

Polarity

positive

Condition

catena

evidence

—

Identifier

46c433109e945de4f2ff86c9f8b0b44d5251ba466d2758cd17c8a22a88cbdfdc

Type

rule_application

Premises

db438b0b14fb7a6e300fd430228baeaecc860d5218c2230d48b225aa90f2ca02
fact-1
fact-3
fact-4

Rule

CatenaGradus

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

assertion · urn:proof:assert:urn:mcp:case#fact-5

attributes

assertion: fact-5

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
3
Type
value
type
name
Integer
value
1

Type

literal

Polarity

positive

Condition

residuum

evidence

—

Identifier

fact-5

Type

assertion

Premises

—

sourceAnchors

—

rule_application · urn:proof:apply:CatenaGradus:280bbd74510474f14bfacb0d3477febddabc4b1587dffdaf1b439ebba3b55589

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
3

Type

literal

Polarity

positive

Condition

catena

evidence

—

Identifier

280bbd74510474f14bfacb0d3477febddabc4b1587dffdaf1b439ebba3b55589

Type

rule_application

Premises

b9cec46229dfe374c4a6a89507e81270c9d73a63a4f430173853872396334111
fact-1
fact-4
fact-5

Rule

CatenaGradus

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

assertion · urn:proof:assert:urn:mcp:case#fact-6

attributes

assertion: fact-6

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
6
Type
value
type
name
Integer
value
1

Type

literal

Polarity

positive

Condition

residuum

evidence

—

Identifier

fact-6

Type

assertion

Premises

—

sourceAnchors

—

rule_application · urn:proof:apply:CatenaDuplicata:b960dcb5f436bbdfbae111bcdf8c237eba6d3c8bffa9480c30f3767c1a5be1bc

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref
Type
value
type
name
Integer
value
6

Type

literal

Polarity

positive

Condition

catena

evidence

—

Identifier

b960dcb5f436bbdfbae111bcdf8c237eba6d3c8bffa9480c30f3767c1a5be1bc

Type

rule_application

Premises

280bbd74510474f14bfacb0d3477febddabc4b1587dffdaf1b439ebba3b55589
fact-1
fact-5
fact-6

Rule

CatenaDuplicata

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

rule_application · urn:proof:apply:ProbatioFermatianaTransacta:95048c69bffee3439284129bba12f3989d11ecef6fd95eabca9572065c0a75d8

attributes

—

conclusion

Arguments

Identifier
probatio
Type
entity_ref

Type

literal

Polarity

positive

Condition

probatio_fermatiana_transacta

evidence

—

Identifier

95048c69bffee3439284129bba12f3989d11ecef6fd95eabca9572065c0a75d8

Type

rule_application

Premises

b960dcb5f436bbdfbae111bcdf8c237eba6d3c8bffa9480c30f3767c1a5be1bc
fact-1
fact-6

Rule

ProbatioFermatianaTransacta

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

rule_application · urn:proof:apply:PrimusPraesumptus/R1:0032244643f2c291d0242e55d164389598889599f5577d46bed6c6ed08e9f1d0

attributes

strength: defeasible

conclusion

Arguments

Type
value
type
name
Integer
value
7

Type

literal

Polarity

positive

Condition

primus

evidence

—

Identifier

0032244643f2c291d0242e55d164389598889599f5577d46bed6c6ed08e9f1d0

Type

rule_application

Premises

95048c69bffee3439284129bba12f3989d11ecef6fd95eabca9572065c0a75d8
fact-1

Rule

PrimusPraesumptus/R1

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

Type

value

type

name: Integer

value

Type

value

type

name: Integer

value

candidate_closure · urn:proof:closure:0d7aeac895165af16cabe73790f8e6122d509d3d81a6b97f131edd487761ad9a

attributes

allApplicableCandidateIds: 0032244643f2c291d0242e55d164389598889599f5577d46bed6c6ed08e9f1d0
conflictKey: {"args":[{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":7}],"predicate":"urn:la:gauss:clir:theorema-fermatianum#primus"}
evaluationInputSemanticHash: sha256:c0ebafc9508a878bc4720fafa551848fbb32b28b4fa7ce5f72a869c4a50c37f1
priorityGraphHash: sha256:c195600c486f311c1d8a40a1bf48943d06189a3877e3473b2deea14afd62b81b
programSemanticHash: sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810
stratum: 0
survivingCandidateIds: 0032244643f2c291d0242e55d164389598889599f5577d46bed6c6ed08e9f1d0

conclusion

Arguments

Type
value
type
name
Integer
value
7

Type

literal

Polarity

positive

Condition

primus

evidence

—

Identifier

0d7aeac895165af16cabe73790f8e6122d509d3d81a6b97f131edd487761ad9a

Type

candidate_closure

Premises

0032244643f2c291d0242e55d164389598889599f5577d46bed6c6ed08e9f1d0

sourceAnchors

—

constraint_check · urn:proof:constraint:TheoremaFermatianumSemperTenet:90ca33fd1ea4783d6a9f1e0ab1655f623cd0f978c942f01e3e69266aac4b6a41

attributes

—

conclusion

constraint: TheoremaFermatianumSemperTenet
requirementStatus: Not established
Calculation status: Undetermined
triggerStatus: Satisfied

evidence

—

Identifier

90ca33fd1ea4783d6a9f1e0ab1655f623cd0f978c942f01e3e69266aac4b6a41

Type

constraint_check

Premises

f12a3f0ef43d198f93d07f0ce6e6d997729b416e0ae1f6b37ab109cca800cd4f

sourceAnchors

—

substitution

Identifier: probatio
Type: entity_ref

query_evaluation · urn:proof:query:mcp

attributes

—

conclusion

literal

Arguments

Type
value
type
name
Integer
value
7

Type

literal

Polarity

positive

Condition

primus

truthStatus

Established

evidence

—

Identifier

mcp

Type

query_evaluation

Premises

0032244643f2c291d0242e55d164389598889599f5577d46bed6c6ed08e9f1d0

sourceAnchors

—

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

{
  "derived": [
    "casus_propositus(urn:case:flt:probatio)",
    "catena(urn:case:flt:probatio, 1)",
    "catena(urn:case:flt:probatio, 2)",
    "catena(urn:case:flt:probatio, 3)",
    "catena(urn:case:flt:probatio, 6)",
    "probatio_fermatiana_transacta(urn:case:flt:probatio)",
    "primus(7)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
    "proofGraph": {
      "nodes": [
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-1"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:flt:probatio",
                "kind": "entity_ref"
              },
              {
                "kind": "value",
                "type": {
                  "name": "urn:law:std#Integer"
                },
                "value": 7
              },
              {
                "kind": "value",
                "type": {
                  "name": "urn:law:std#Integer"
                },
                "value": 2
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:la:gauss:clir:theorema-fermatianum#propositum"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-1",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:flt:probatio",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:la:gauss:clir:theorema-fermatianum#casus_propositus"
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Article 46

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

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      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
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Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

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Full engine response

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Execution · JSON

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Download JSON ↓

Display metadata

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JSON · calculations, sources and exact data

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evaluation SHA-256: sha256:a1ae8ee430e1d60aa4879e8b8cccc09a868786933ea2f1aae47e23443562b07b

Original data · JSON

{
  "args": [
    7
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        7,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        6
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
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        "urn:case:flt:probatio",
        2,
        4
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      "predicate": "residuum"
    },
    {
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        "urn:case:flt:probatio",
        3,
        1
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      "predicate": "residuum"
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    {
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        "urn:case:flt:probatio",
        6,
        1
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    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "primus",
  "proof": true
}

Опорный случай: для настоящего простого числа критерий даёт верный ответ.

Condition

341 проходит тест по основанию 2

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

probatio Fermatiana transacta: potestas p-1 unitati congrua

probatio

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 341basis: 2
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 340
residuum minimum potestatis exponentis dati
exponens valor
1 2
2 4
4 16
5 32
10 1
20 1
21 2
42 4
84 16
85 32
170 1
340 1

exponens	valor
1	2
2	4
4	16
5	32
10	1
20	1
21	2
42	4
84	16
85	32
170	1
340	1

Additional details

Include proof: Yes

Original data · JSON

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  ],
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        2
      ],
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    },
    {
      "args": [
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        340
      ],
      "predicate": "vestigium"
    },
    {
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        1,
        2
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      "predicate": "residuum"
    },
    {
      "args": [
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        2,
        4
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    },
    {
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        4,
        16
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    },
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        5,
        32
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      "predicate": "residuum"
    },
    {
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        10,
        1
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      "predicate": "residuum"
    },
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        20,
        1
      ],
      "predicate": "residuum"
    },
    {
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        21,
        2
      ],
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    },
    {
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        42,
        4
      ],
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    },
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        84,
        16
      ],
      "predicate": "residuum"
    },
    {
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        85,
        32
      ],
      "predicate": "residuum"
    },
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        170,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340,
        1
      ],
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    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
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}

Why this resultApplied rules and conditions

Derivation path27 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 341; basis: 2
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 42; valor: 4
case fact
3
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 84; valor: 16
case fact
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 85; valor: 32
case fact
5
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 170; valor: 1
case fact
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 340; valor: 1
case fact
7
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 2
case fact
8
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
9
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 4
case fact
10
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
11
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 4; valor: 16
case fact
12
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 4
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
13
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 5; valor: 32
case fact
14
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 5
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
15
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 10; valor: 1
case fact
16
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 10
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
17
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 20; valor: 1
case fact
18
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 20
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
19
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 21; valor: 2
case fact
20
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 21
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
21
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 42
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
22
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 84
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
23
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 85
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
24
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 170
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
25
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 340
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
26
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
27
Query evaluation
query

verified by the engine: 14 · case fact: 13 · Full graph: 32 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation2

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1

Derived result for this query

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr: probatio

Other derived facts13

probatio
pr: probatio
Subject shared by the facts below
casus propositus
catena potestatum ad exponentem usque probata
exponens
1
2
4
5
10
20
21
42
84
85
170
340

exponens
1
2
4
5
10
20
21
42
84
85
170
340

casus propositus
pr
probatio

catena potestatum ad exponentem usque probata
pr	exponens
probatio	1
probatio	2
probatio	4
probatio	5
probatio	10
probatio	20
probatio	21
probatio	42
probatio	84
probatio	85
probatio	170
probatio	340

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr
probatio

1 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet341Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 14 layer

Proof nodes: 32 · assertion 13, rule_application 16, candidate_closure 1, constraint_check 1, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

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SourcesExcerpts: 2

Article 46

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

Original data · JSON

{
  "contentHash": "sha256:9d4fd73665d88ea62e99bdb067f2d4791ac28e3ce47b5722ebe0d8406d0e5527",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
  "kind": "fragment",
  "locator": "article/46",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:3d8481c723da99a966d5a5240c0761a80446076ff8d5e2751cca833ee3027e13",
      "language": "la",
      "status": "official",
      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
    }
  ]
}

Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

Original data · JSON

{
  "contentHash": "sha256:bada760fcd34e13cef68ac9ef2a8343055844742af9259bcf3c8b58181320c1b",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART50",
  "kind": "fragment",
  "locator": "article/50",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:9a2d7f1a9f694545a613962cadc9694bd27b4436d947b39fcbd30251a3b5fcdf",
      "language": "la",
      "status": "official",
      "text": "Fermatii Theorema.\n\nQuum igitur $\\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem\ncongruentiae $a^{t} \\equiv 1$ ad potestatem exponentis $\\tfrac{p-1}{t}$, $a^{p-1} \\equiv 1$,\nsive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens.\n\nTheorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem\nomni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid.\nFermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor\nnon adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus\ndemonstrationem publici iuris fecit, in diss. cui titulus\nTheorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop.  T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p.  106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia\n$1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769\np. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena\nesse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam\nsuperaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad\nalias investigationes infra adhibebitur."
    }
  ]
}

Packages in the snapshot

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Technical dataFull response, parameters and checksums

Calculation status: COMPUTED

Full engine response

probatio_fermatiana_transacta: TRUE_ONLY — установлено Выведено правом: casus_propositus(urn:case:flt:probatio); catena(urn:case:flt:probatio, 1); catena(urn:case:flt:probatio, 2); catena(urn:case:flt:probatio, 4); catena(urn:case:flt:probatio, 5); catena(urn:case:flt:probatio, 10); catena(urn:case:flt:probatio, 20); catena(urn:case:flt:probatio, 21); catena(urn:case:flt:probatio, 42); catena(urn:case:flt:probatio, 84); catena(urn:case:flt:probatio, 85); catena(urn:case:flt:probatio, 170); catena(urn:case:flt:probatio, 340); probatio_fermatiana_transacta(urn:case:flt:probatio) …и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta Ответ поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(341) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…) proof-граф: 32 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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Execution · JSON

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Display metadata

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JSON · calculations, sources and exact data

{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 6,
      "fragments": [
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART50"
      ],
      "jurisdiction": "none",
      "namespace": "urn:la:gauss:clir:theorema-fermatianum",
      "package": "la-gauss-theorema-fermatianum",
      "title": "Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1"
    }
  ],
  "caseHash": "sha256:5ce6fd5648e2dccccb8defb6a73decffe425fe4011b6068d13e4d327ed5044af",
  "codeHash": "sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699",
  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810",
  "resultHash": "sha256:185a964681ac6fca7c08c6602ef9c243c37d4124229b788cfffdb6857c8a6cdf",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}

evaluation SHA-256: sha256:d2d8dac3bf00025a09151ba20f705ada8f84ccbf9efaadd2483e7c3c23f67252

Original data · JSON

{
  "args": [
    "urn:case:flt:probatio"
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        341,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        5,
        32
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        10,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        20,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        21,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        42,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        84,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        85,
        32
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        170,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
  "proof": true
}

След степени 340 по модулю 341 сходится к единице: тест пройден, хотя 341 = 11 · 31.

Condition

341 объявлено простым без делителя

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

numerus primus

341

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 341basis: 2
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 340
residuum minimum potestatis exponentis dati
exponens valor
1 2
2 4
4 16
5 32
10 1
20 1
21 2
42 4
84 16
85 32
170 1
340 1

exponens	valor
1	2
2	4
4	16
5	32
10	1
20	1
21	2
42	4
84	16
85	32
170	1
340	1

Additional details

Include proof: Yes

Original data · JSON

{
  "args": [
    341
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        341,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        5,
        32
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        10,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        20,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        21,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        42,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        84,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        85,
        32
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        170,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "primus",
  "proof": true
}

Why this resultApplied rules and conditions

Derivation path28 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 341; basis: 2
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 42; valor: 4
case fact
3
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 84; valor: 16
case fact
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 85; valor: 32
case fact
5
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 170; valor: 1
case fact
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 340; valor: 1
case fact
7
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 2
case fact
8
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
9
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 4
case fact
10
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
11
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 4; valor: 16
case fact
12
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 4
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
13
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 5; valor: 32
case fact
14
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 5
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
15
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 10; valor: 1
case fact
16
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 10
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
17
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 20; valor: 1
case fact
18
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 20
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
19
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 21; valor: 2
case fact
20
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 21
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
21
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 42
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
22
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 84
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
23
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 85
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
24
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 170
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
25
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 340
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
26
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
27
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
numerus primus: modulus: 341
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1
rule
28
Query evaluation
query

verified by the engine: 15 · case fact: 13 · Full graph: 32 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation1

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus

Derived result for this query

numerus primus
modulus: 341

numerus primus
modulus
341

14 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet341Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 15 layer

Proof nodes: 32 · assertion 13, rule_application 16, candidate_closure 1, constraint_check 1, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

SourcesExcerpts: 2

Article 46

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

Original data · JSON

{
  "contentHash": "sha256:9d4fd73665d88ea62e99bdb067f2d4791ac28e3ce47b5722ebe0d8406d0e5527",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
  "kind": "fragment",
  "locator": "article/46",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:3d8481c723da99a966d5a5240c0761a80446076ff8d5e2751cca833ee3027e13",
      "language": "la",
      "status": "official",
      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
    }
  ]
}

Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

Original data · JSON

{
  "contentHash": "sha256:bada760fcd34e13cef68ac9ef2a8343055844742af9259bcf3c8b58181320c1b",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART50",
  "kind": "fragment",
  "locator": "article/50",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:9a2d7f1a9f694545a613962cadc9694bd27b4436d947b39fcbd30251a3b5fcdf",
      "language": "la",
      "status": "official",
      "text": "Fermatii Theorema.\n\nQuum igitur $\\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem\ncongruentiae $a^{t} \\equiv 1$ ad potestatem exponentis $\\tfrac{p-1}{t}$, $a^{p-1} \\equiv 1$,\nsive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens.\n\nTheorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem\nomni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid.\nFermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor\nnon adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus\ndemonstrationem publici iuris fecit, in diss. cui titulus\nTheorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop.  T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p.  106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia\n$1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769\np. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena\nesse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam\nsuperaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad\nalias investigationes infra adhibebitur."
    }
  ]
}

Packages in the snapshot

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Technical dataFull response, parameters and checksums

Calculation status: COMPUTED

Full engine response

primus: TRUE_ONLY — установлено Выведено правом: primus(341) …и ещё 14 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta Ответ поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(341) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…) proof-граф: 32 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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Execution · JSON

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Display metadata

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JSON · calculations, sources and exact data

{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 6,
      "fragments": [
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      "jurisdiction": "none",
      "namespace": "urn:la:gauss:clir:theorema-fermatianum",
      "package": "la-gauss-theorema-fermatianum",
      "title": "Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1"
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  "caseHash": "sha256:5ce6fd5648e2dccccb8defb6a73decffe425fe4011b6068d13e4d327ed5044af",
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  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810",
  "resultHash": "sha256:b583563cb6f94007f483c3c3b40f5d3135f30efb90823000dfb32d6d1341f187",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}

evaluation SHA-256: sha256:7ff97d9681ca8a74f2bd211f9ca7d5035f840fce29bb040727696c9dc75d2f60

Original data · JSON

{
  "args": [
    341
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        341,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
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        2,
        4
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      "predicate": "residuum"
    },
    {
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        16
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      "predicate": "residuum"
    },
    {
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        5,
        32
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      "predicate": "residuum"
    },
    {
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        10,
        1
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      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        20,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        21,
        2
      ],
      "predicate": "residuum"
    },
    {
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        42,
        4
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      "predicate": "residuum"
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    {
      "args": [
        "urn:case:flt:probatio",
        84,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        85,
        32
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        170,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        340,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "primus",
  "proof": true
}

Делитель не предъявлен, критерий пройден — вывод «простое» следует из посылок, и в этом слабость теста.

Condition

561 по основанию 2

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

probatio Fermatiana transacta: potestas p-1 unitati congrua

probatio

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 561basis: 2
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 560
residuum minimum potestatis exponentis dati
exponens valor
1 2
2 4
4 16
8 256
16 460
17 359
34 412
35 263
70 166
140 67
280 1
560 1

exponens	valor
1	2
2	4
4	16
8	256
16	460
17	359
34	412
35	263
70	166
140	67
280	1
560	1

Additional details

Include proof: Yes

Original data · JSON

{
  "args": [
    "urn:case:flt:probatio"
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        561,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        8,
        256
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        16,
        460
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        17,
        359
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        34,
        412
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        35,
        263
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        70,
        166
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        140,
        67
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        280,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
  "proof": true
}

Why this resultApplied rules and conditions

Derivation path27 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 561; basis: 2
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 35; valor: 263
case fact
3
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 70; valor: 166
case fact
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 140; valor: 67
case fact
5
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 280; valor: 1
case fact
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 560; valor: 1
case fact
7
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 2
case fact
8
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
9
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 4
case fact
10
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
11
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 4; valor: 16
case fact
12
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 4
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
13
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 8; valor: 256
case fact
14
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 8
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
15
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 16; valor: 460
case fact
16
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 16
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
17
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 17; valor: 359
case fact
18
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 17
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
19
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 34; valor: 412
case fact
20
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 34
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
21
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 35
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
22
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 70
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
23
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 140
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
24
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 280
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
25
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 560
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
26
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
27
Query evaluation
query

verified by the engine: 14 · case fact: 13 · Full graph: 32 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation2

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1

Derived result for this query

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr: probatio

Other derived facts13

probatio
pr: probatio
Subject shared by the facts below
casus propositus
catena potestatum ad exponentem usque probata
exponens
1
2
4
8
16
17
34
35
70
140
280
560

exponens
1
2
4
8
16
17
34
35
70
140
280
560

casus propositus
pr
probatio

catena potestatum ad exponentem usque probata
pr	exponens
probatio	1
probatio	2
probatio	4
probatio	8
probatio	16
probatio	17
probatio	34
probatio	35
probatio	70
probatio	140
probatio	280
probatio	560

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr
probatio

1 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet561Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 14 layer

Proof nodes: 32 · assertion 13, rule_application 16, candidate_closure 1, constraint_check 1, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

SourcesExcerpts: 2

Article 46

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

Original data · JSON

{
  "contentHash": "sha256:9d4fd73665d88ea62e99bdb067f2d4791ac28e3ce47b5722ebe0d8406d0e5527",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
  "kind": "fragment",
  "locator": "article/46",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:3d8481c723da99a966d5a5240c0761a80446076ff8d5e2751cca833ee3027e13",
      "language": "la",
      "status": "official",
      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
    }
  ]
}

Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

Original data · JSON

{
  "contentHash": "sha256:bada760fcd34e13cef68ac9ef2a8343055844742af9259bcf3c8b58181320c1b",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART50",
  "kind": "fragment",
  "locator": "article/50",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:9a2d7f1a9f694545a613962cadc9694bd27b4436d947b39fcbd30251a3b5fcdf",
      "language": "la",
      "status": "official",
      "text": "Fermatii Theorema.\n\nQuum igitur $\\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem\ncongruentiae $a^{t} \\equiv 1$ ad potestatem exponentis $\\tfrac{p-1}{t}$, $a^{p-1} \\equiv 1$,\nsive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens.\n\nTheorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem\nomni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid.\nFermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor\nnon adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus\ndemonstrationem publici iuris fecit, in diss. cui titulus\nTheorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop.  T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p.  106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia\n$1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769\np. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena\nesse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam\nsuperaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad\nalias investigationes infra adhibebitur."
    }
  ]
}

Packages in the snapshot

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Technical dataFull response, parameters and checksums

Calculation status: COMPUTED

Full engine response

probatio_fermatiana_transacta: TRUE_ONLY — установлено Выведено правом: casus_propositus(urn:case:flt:probatio); catena(urn:case:flt:probatio, 1); catena(urn:case:flt:probatio, 2); catena(urn:case:flt:probatio, 4); catena(urn:case:flt:probatio, 8); catena(urn:case:flt:probatio, 16); catena(urn:case:flt:probatio, 17); catena(urn:case:flt:probatio, 34); catena(urn:case:flt:probatio, 35); catena(urn:case:flt:probatio, 70); catena(urn:case:flt:probatio, 140); catena(urn:case:flt:probatio, 280); catena(urn:case:flt:probatio, 560); probatio_fermatiana_transacta(urn:case:flt:probatio) …и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta Ответ поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…) proof-граф: 32 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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Execution · JSON

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Display metadata

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JSON · calculations, sources and exact data

{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 6,
      "fragments": [
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART50"
      ],
      "jurisdiction": "none",
      "namespace": "urn:la:gauss:clir:theorema-fermatianum",
      "package": "la-gauss-theorema-fermatianum",
      "title": "Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1"
    }
  ],
  "caseHash": "sha256:e2303d459d2bcf33d6303e3cc9ba80f3caa4d690f39f348bf6193f82cdfcadc6",
  "codeHash": "sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699",
  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810",
  "resultHash": "sha256:c7223e986ec73c6222fc9599d7e9f55bb32702126d0f70c2bc69f412661fb2f8",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}

evaluation SHA-256: sha256:ef7b1de8d34229e7f81e7372412849bb0a7f36d52320d421f2ca77e219aa0c58

Original data · JSON

{
  "args": [
    "urn:case:flt:probatio"
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        561,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        8,
        256
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        16,
        460
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        17,
        359
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        34,
        412
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        35,
        263
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        70,
        166
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        140,
        67
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        280,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
  "proof": true
}

Первое основание тест не различает.

Condition

561 по основанию 5

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

probatio Fermatiana transacta: potestas p-1 unitati congrua

probatio

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 561basis: 5
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 560
residuum minimum potestatis exponentis dati
exponens valor
1 5
2 25
4 64
8 169
16 511
17 311
34 229
35 23
70 529
140 463
280 67
560 1

exponens	valor
1	5
2	25
4	64
8	169
16	511
17	311
34	229
35	23
70	529
140	463
280	67
560	1

Additional details

Include proof: Yes

Original data · JSON

{
  "args": [
    "urn:case:flt:probatio"
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        561,
        5
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        5
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        25
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        64
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        8,
        169
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        16,
        511
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        17,
        311
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        34,
        229
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        35,
        23
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        70,
        529
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        140,
        463
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        280,
        67
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
  "proof": true
}

Why this resultApplied rules and conditions

Derivation path27 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 561; basis: 5
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 35; valor: 23
case fact
3
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 70; valor: 529
case fact
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 140; valor: 463
case fact
5
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 280; valor: 67
case fact
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 560; valor: 1
case fact
7
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 5
case fact
8
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
9
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 25
case fact
10
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
11
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 4; valor: 64
case fact
12
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 4
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
13
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 8; valor: 169
case fact
14
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 8
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
15
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 16; valor: 511
case fact
16
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 16
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
17
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 17; valor: 311
case fact
18
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 17
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
19
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 34; valor: 229
case fact
20
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 34
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
21
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 35
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
22
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 70
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
23
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 140
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
24
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 280
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
25
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 560
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
26
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
27
Query evaluation
query

verified by the engine: 14 · case fact: 13 · Full graph: 32 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation2

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1

Derived result for this query

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr: probatio

Other derived facts13

probatio
pr: probatio
Subject shared by the facts below
casus propositus
catena potestatum ad exponentem usque probata
exponens
1
2
4
8
16
17
34
35
70
140
280
560

exponens
1
2
4
8
16
17
34
35
70
140
280
560

casus propositus
pr
probatio

catena potestatum ad exponentem usque probata
pr	exponens
probatio	1
probatio	2
probatio	4
probatio	8
probatio	16
probatio	17
probatio	34
probatio	35
probatio	70
probatio	140
probatio	280
probatio	560

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr
probatio

1 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet561Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 14 layer

Proof nodes: 32 · assertion 13, rule_application 16, candidate_closure 1, constraint_check 1, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

SourcesExcerpts: 2

Article 46

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

Original data · JSON

{
  "contentHash": "sha256:9d4fd73665d88ea62e99bdb067f2d4791ac28e3ce47b5722ebe0d8406d0e5527",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
  "kind": "fragment",
  "locator": "article/46",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:3d8481c723da99a966d5a5240c0761a80446076ff8d5e2751cca833ee3027e13",
      "language": "la",
      "status": "official",
      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
    }
  ]
}

Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

Original data · JSON

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  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART50",
  "kind": "fragment",
  "locator": "article/50",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:9a2d7f1a9f694545a613962cadc9694bd27b4436d947b39fcbd30251a3b5fcdf",
      "language": "la",
      "status": "official",
      "text": "Fermatii Theorema.\n\nQuum igitur $\\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem\ncongruentiae $a^{t} \\equiv 1$ ad potestatem exponentis $\\tfrac{p-1}{t}$, $a^{p-1} \\equiv 1$,\nsive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens.\n\nTheorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem\nomni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid.\nFermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor\nnon adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus\ndemonstrationem publici iuris fecit, in diss. cui titulus\nTheorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop.  T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p.  106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia\n$1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769\np. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena\nesse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam\nsuperaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad\nalias investigationes infra adhibebitur."
    }
  ]
}

Packages in the snapshot

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Technical dataFull response, parameters and checksums

Calculation status: COMPUTED

Full engine response

probatio_fermatiana_transacta: TRUE_ONLY — установлено Выведено правом: casus_propositus(urn:case:flt:probatio); catena(urn:case:flt:probatio, 1); catena(urn:case:flt:probatio, 2); catena(urn:case:flt:probatio, 4); catena(urn:case:flt:probatio, 8); catena(urn:case:flt:probatio, 16); catena(urn:case:flt:probatio, 17); catena(urn:case:flt:probatio, 34); catena(urn:case:flt:probatio, 35); catena(urn:case:flt:probatio, 70); catena(urn:case:flt:probatio, 140); catena(urn:case:flt:probatio, 280); catena(urn:case:flt:probatio, 560); probatio_fermatiana_transacta(urn:case:flt:probatio) …и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta Ответ поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…) proof-граф: 32 узлов — поле evaluation готово для law_explain

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JSON · calculations, sources and exact data

{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 6,
      "fragments": [
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
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      ],
      "jurisdiction": "none",
      "namespace": "urn:la:gauss:clir:theorema-fermatianum",
      "package": "la-gauss-theorema-fermatianum",
      "title": "Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1"
    }
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  "resultHash": "sha256:899a16675679ae3bc6a25bedba834acf9fe5cfbacb9ecec2c96704e7bcb88173",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}

evaluation SHA-256: sha256:41f66af547c294c452ae931339d79f58978ca0c9785f383beb814d03f4a6edf1

Original data · JSON

{
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    "urn:case:flt:probatio"
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        561,
        5
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        5
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        25
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        64
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        8,
        169
      ],
      "predicate": "residuum"
    },
    {
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        "urn:case:flt:probatio",
        16,
        511
      ],
      "predicate": "residuum"
    },
    {
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        "urn:case:flt:probatio",
        17,
        311
      ],
      "predicate": "residuum"
    },
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        "urn:case:flt:probatio",
        34,
        229
      ],
      "predicate": "residuum"
    },
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        35,
        23
      ],
      "predicate": "residuum"
    },
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        70,
        529
      ],
      "predicate": "residuum"
    },
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        "urn:case:flt:probatio",
        140,
        463
      ],
      "predicate": "residuum"
    },
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      "args": [
        "urn:case:flt:probatio",
        280,
        67
      ],
      "predicate": "residuum"
    },
    {
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        "urn:case:flt:probatio",
        560,
        1
      ],
      "predicate": "residuum"
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  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
  "proof": true
}

Второе основание тоже проходит: у обычного псевдопростого здесь была бы осечка.

Condition

561 по основанию 7

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

probatio Fermatiana transacta: potestas p-1 unitati congrua

probatio

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 561basis: 7
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 560
residuum minimum potestatis exponentis dati
exponens valor
1 7
2 49
4 157
8 526
16 103
17 160
34 355
35 241
70 298
140 166
280 67
560 1

exponens	valor
1	7
2	49
4	157
8	526
16	103
17	160
34	355
35	241
70	298
140	166
280	67
560	1

Additional details

Include proof: Yes

Original data · JSON

{
  "args": [
    "urn:case:flt:probatio"
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        561,
        7
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        7
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        49
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        157
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        8,
        526
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        16,
        103
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        17,
        160
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        34,
        355
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        35,
        241
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        70,
        298
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        140,
        166
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        280,
        67
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560,
        1
      ],
      "predicate": "residuum"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "probatio_fermatiana_transacta",
  "proof": true
}

Why this resultApplied rules and conditions

Derivation path27 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 561; basis: 7
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 35; valor: 241
case fact
3
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 70; valor: 298
case fact
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 140; valor: 166
case fact
5
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 280; valor: 67
case fact
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 560; valor: 1
case fact
7
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 7
case fact
8
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
9
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 49
case fact
10
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
11
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 4; valor: 157
case fact
12
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 4
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
13
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 8; valor: 526
case fact
14
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 8
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
15
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 16; valor: 103
case fact
16
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 16
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
17
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 17; valor: 160
case fact
18
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 17
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
19
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 34; valor: 355
case fact
20
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 34
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
21
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 35
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
22
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 70
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
23
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 140
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
24
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 280
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
25
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 560
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
26
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
27
Query evaluation
query

verified by the engine: 14 · case fact: 13 · Full graph: 32 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation2

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1

Derived result for this query

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr: probatio

Other derived facts13

probatio
pr: probatio
Subject shared by the facts below
casus propositus
catena potestatum ad exponentem usque probata
exponens
1
2
4
8
16
17
34
35
70
140
280
560

exponens
1
2
4
8
16
17
34
35
70
140
280
560

casus propositus
pr
probatio

catena potestatum ad exponentem usque probata
pr	exponens
probatio	1
probatio	2
probatio	4
probatio	8
probatio	16
probatio	17
probatio	34
probatio	35
probatio	70
probatio	140
probatio	280
probatio	560

probatio Fermatiana transacta: potestas p-1 unitati congrua
pr
probatio

1 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet561Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 14 layer

Proof nodes: 32 · assertion 13, rule_application 16, candidate_closure 1, constraint_check 1, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

SourcesExcerpts: 2

Article 46

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

Original data · JSON

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  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
  "kind": "fragment",
  "locator": "article/46",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
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      "language": "la",
      "status": "official",
      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
    }
  ]
}

Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

Original data · JSON

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      "text": "Fermatii Theorema.\n\nQuum igitur $\\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem\ncongruentiae $a^{t} \\equiv 1$ ad potestatem exponentis $\\tfrac{p-1}{t}$, $a^{p-1} \\equiv 1$,\nsive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens.\n\nTheorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem\nomni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid.\nFermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor\nnon adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus\ndemonstrationem publici iuris fecit, in diss. cui titulus\nTheorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop.  T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p.  106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia\n$1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769\np. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena\nesse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam\nsuperaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad\nalias investigationes infra adhibebitur."
    }
  ]
}

Packages in the snapshot

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Technical dataFull response, parameters and checksums

Calculation status: COMPUTED

Full engine response

probatio_fermatiana_transacta: TRUE_ONLY — установлено Выведено правом: casus_propositus(urn:case:flt:probatio); catena(urn:case:flt:probatio, 1); catena(urn:case:flt:probatio, 2); catena(urn:case:flt:probatio, 4); catena(urn:case:flt:probatio, 8); catena(urn:case:flt:probatio, 16); catena(urn:case:flt:probatio, 17); catena(urn:case:flt:probatio, 34); catena(urn:case:flt:probatio, 35); catena(urn:case:flt:probatio, 70); catena(urn:case:flt:probatio, 140); catena(urn:case:flt:probatio, 280); catena(urn:case:flt:probatio, 560); probatio_fermatiana_transacta(urn:case:flt:probatio) …и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta Ответ поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…) proof-граф: 32 узлов — поле evaluation готово для law_explain

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JSON · calculations, sources and exact data

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evaluation SHA-256: sha256:d50d4d9625c9ba8e2caa7659892913d34f45b102c036eb61ffd11a3330e0e20a

Original data · JSON

{
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        561,
        7
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И третье. Число Кармайкла обманывает все взаимно простые основания сразу.

Condition

561 объявлено простым

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

numerus primus

561

Input facts

probatio
pr: probatio
Subject shared by the facts below
progressio geometrica secundum modulum p ad basim a
modulus: 561basis: 2
vestigium potestatum ab exponente uno usque ad longitudinem datam
longitudo: 560
residuum minimum potestatis exponentis dati
exponens valor
1 2
2 4
4 16
8 256
16 460
17 359
34 412
35 263
70 166
140 67
280 1
560 1
divisor allatus qui modulum metitur
divisor: 4

exponens	valor
1	2
2	4
4	16
8	256
16	460
17	359
34	412
35	263
70	166
140	67
280	1
560	1

Additional details

Include proof: Yes

Original data · JSON

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}

Why this resultApplied rules and conditions

Derivation path28 steps

1
progressio geometrica secundum modulum p ad basim a
pr: urn:case:flt:probatio; modulus: 561; basis: 2
case fact
2
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 35; valor: 263
case fact
3
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 70; valor: 166
case fact
4
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 140; valor: 67
case fact
5
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 280; valor: 1
case fact
6
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 560; valor: 1
case fact
7
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 1; valor: 2
case fact
8
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 1
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
rule
9
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 2; valor: 4
case fact
10
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 2
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
11
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 4; valor: 16
case fact
12
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 4
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
13
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 8; valor: 256
case fact
14
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 8
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
15
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 16; valor: 460
case fact
16
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 16
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
17
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 17; valor: 359
case fact
18
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 17
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
19
residuum minimum potestatis exponentis dati
pr: urn:case:flt:probatio; exponens: 34; valor: 412
case fact
20
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 34
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
21
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 35
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
rule
22
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 70
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
23
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 140
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
24
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 280
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
25
donec ad terminum $a^{(2t)}$ perveniatur
catena potestatum ad exponentem usque probata: pr: urn:case:flt:probatio; exponens: 560
art. 46
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
rule
26
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
probatio Fermatiana transacta: potestas p-1 unitati congrua: pr: urn:case:flt:probatio
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta
rule
27
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
numerus primus: modulus: 561
art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1
rule
28
Query evaluation
query

verified by the engine: 15 · case fact: 13 · Full graph: 32 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

donec ad terminum $a^{(2t)}$ perveniatur
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaDuplicata
Scilicet si $a^t$ est unitati congruum, erit $a^{(t+1)}$ congruum ipsi $a$
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaGradus
In omni progressione geometrica $1, a, aa, a^3$ etc. praeter primum 1
Identifier
urn:la:gauss:clir:theorema-fermatianum#CatenaInitium
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1
sive $a^{(p-1)}-1$ semper per $p$ divisibilis est, quando p est primus ipsum a non metiens
Identifier
urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta

Other rules in the evaluation1

Applied in the overall evaluation, but not on the proof path for this answer.

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина

casus propositus: modulus et basis dati
Identifier
urn:la:gauss:clir:theorema-fermatianum#CasusPropositus

Derived result for this query

numerus primus
modulus: 561

numerus primus
modulus
561

14 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

1
numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur
What is missing
- numerus compositus: divisorem habet561Not establishedthis is the missing one
Source: art. 50
Identifier
urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2
rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 15 layer

Proof nodes: 32 · assertion 13, rule_application 16, candidate_closure 1, constraint_check 1, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

Calendar and proof identifiers

Proof reference: mcp

Original reasoning · JSON

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SourcesExcerpts: 2

Article 46

Quando progressio ultra terminum, qui unitati est congruus, continuatur, eadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \equiv 1$, erit $a^{t+1} \equiv a$, $a^{t+2} \equiv aa$ etc., donec ad terminum $a^{2t}\!$ perveniatur, cuius residuum minimum iterum erit $\equiv 1$, atque residuorum periodum denuo inchoat. Habetur itaque periodus $t$ residua comprehendens, quae simulac finita est ab initio semper repetitur; neque alia residua quam quae in hac periodo continentur, in tota progressione occurrere possunt. Generaliter erit $a^{mt} \equiv 1$, et $a^{mt+n} \equiv a^{n}\!$, id quod per designationem nostram ita exhibetur: Si $\textstyle r \equiv \rho \pmod{t}$, erit $\textstyle a^{r} \equiv a^{\rho} \pmod{t}$.

Original data · JSON

{
  "contentHash": "sha256:9d4fd73665d88ea62e99bdb067f2d4791ac28e3ce47b5722ebe0d8406d0e5527",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
  "kind": "fragment",
  "locator": "article/46",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:3d8481c723da99a966d5a5240c0761a80446076ff8d5e2751cca833ee3027e13",
      "language": "la",
      "status": "official",
      "text": "Quando progressio ultra terminum, qui unitati est congruus, continuatur,\neadem, quae ab initio habebantur, residua prodeunt iterum. Scilicet si $a^{t} \\equiv 1$, erit $a^{t+1} \\equiv a$, $a^{t+2} \\equiv aa$ etc., donec ad terminum $a^{2t}\\!$ perveniatur, cuius residuum minimum iterum erit $\\equiv 1$, atque residuorum periodum denuo inchoat. Habetur\nitaque periodus $t$ residua comprehendens, quae simulac finita est ab initio\nsemper repetitur; neque alia residua quam quae in hac periodo continentur, in tota\nprogressione occurrere possunt. Generaliter erit $a^{mt} \\equiv 1$, et $a^{mt+n} \\equiv a^{n}\\!$,\nid quod per designationem nostram ita exhibetur:\n\nSi $\\textstyle r \\equiv \\rho \\pmod{t}$, erit $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."
    }
  ]
}

Article 50

Fermatii Theorema. Quum igitur $\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem congruentiae $a^{t} \equiv 1$ ad potestatem exponentis $\tfrac{p-1}{t}$, $a^{p-1} \equiv 1$, sive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens. Theorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem omni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid. Fermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor non adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus demonstrationem publici iuris fecit, in diss. cui titulus Theorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop. T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p. 106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia $1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769 p. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena esse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam superaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad alias investigationes infra adhibebitur.

Original data · JSON

{
  "contentHash": "sha256:bada760fcd34e13cef68ac9ef2a8343055844742af9259bcf3c8b58181320c1b",
  "edition": "urn:la:gauss:clir:theorema-fermatianum#DISQUISITIONES_LA",
  "fragmentKind": "article",
  "id": "urn:la:gauss:clir:theorema-fermatianum#DA_ART50",
  "kind": "fragment",
  "locator": "article/50",
  "package": "urn:la:gauss:clir:theorema-fermatianum",
  "texts": [
    {
      "contentHash": "sha256:9a2d7f1a9f694545a613962cadc9694bd27b4436d947b39fcbd30251a3b5fcdf",
      "language": "la",
      "status": "official",
      "text": "Fermatii Theorema.\n\nQuum igitur $\\tfrac{p-1}{t}$ sit integer, sequitur evehendo utramque partem\ncongruentiae $a^{t} \\equiv 1$ ad potestatem exponentis $\\tfrac{p-1}{t}$, $a^{p-1} \\equiv 1$,\nsive $a^{p-1}-1$ semper per $p$ divisibilis est, quando $p$ est primus ipsum $a$ non metiens.\n\nTheorema hoc, quod tum propter elegantiam tum propter eximiam utilitatem\nomni attentione dignum, ab inventore theorema Fermatianum appellari solet. Vid.\nFermatii Opera Mathem. Tolosae 1679 fol. p. 163. Demonstrationem inventor\nnon adiecit, quam tamen in potestate sua esse professus est. Ill. Euler primus\ndemonstrationem publici iuris fecit, in diss. cui titulus\nTheorematum quorundam ad numeros primos spectantium demonstratio, Comm. Acad. Petrop.  T. VIII [Nota: In comment. anteriore vir summus ad scopum nondum pervenerat. Comm. Petr. T. VI p. 106. — In controversia famosa inter Maupertuis et König, a principio actionis minimae orta, sed mox ad res heterogeneas egressa, König in manibus se habere dixit autographum Leibnitianum, in quo demonstratio huius theorematis cum Euleriana prorsus conspirans contineatur. Appel au public, p.  106. Licet vero fidem huic testimonio denegare nolimus, certe Leibnitius inventum suum numquam publicavit. Conf. Hist.de l'Ac. de Prusse, A. 1750 p. 530.]. Innititur ista evolutioni potestatis $(a+1)^{p}\\!$, ubi ex coëfficientium forma facillime deducitur, $(a+1)^{p}-a^p-1$ semper per $p$ fore divisibilem, adeoque $(a+1)^{p}-(a+1)$ per $p$ divisibilem fore, quando $a^{p}-a$ per $p$ sit divisibilis. Iam quia\n$1^{p}-1$ semper per $p$ divisibilis est, etiam $2^{p}-2$ semper erit; hinc etiam $3^{p}-3$ etc. generaliterque $a^{p}-a$. Quodsi itaque $p$ ipsum $a$ non metitur, etiam $a^{p-1}-1$ per $p$ divisibilis erit. Haec sufficient ad methodi indolem declarandam. Clar. Lambert similem demonstrationem tradidit in Actis Erudit. 1769\np. 109. Quia vero evolutio potestatis binomii a theoria numerorum satis aliena\nesse videbatur, aliam demonstrationem ill. Euler investigavit, quae exstat Comment. nov. Petr. T. VII p. 70, atque cum ea quam nos art. praec. exposuimus prorsus convenit. In sequentibus adhuc aliae quaedam se nobis offerent. Hoc loco unam\nsuperaddere liceat, quae similibus principiis innititur, uti prima ill. Euleri. Propositio sequens, cuius casus tantum particularis est theorema nostrum, etiam ad\nalias investigationes infra adhibebitur."
    }
  ]
}

Packages in the snapshot

Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1

Technical dataFull response, parameters and checksums

Calculation status: COMPUTED

Full engine response

primus: TRUE_ONLY — установлено Выведено правом: primus(561) …и ещё 14 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta Ответ поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…) proof-граф: 32 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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Execution · JSON

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Display metadata

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JSON · calculations, sources and exact data

{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 6,
      "fragments": [
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART46",
        "urn:la:gauss:clir:theorema-fermatianum#DA_ART50"
      ],
      "jurisdiction": "none",
      "namespace": "urn:la:gauss:clir:theorema-fermatianum",
      "package": "la-gauss-theorema-fermatianum",
      "title": "Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — вне юрисдикции государства — доктрина — EXECUTABLE 6 §33.1"
    }
  ],
  "caseHash": "sha256:61be5f0522b3a0a4811843cf90619914b4c9ed107b8d6f2b37eb664a49db2ab1",
  "codeHash": "sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699",
  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810",
  "resultHash": "sha256:e3cf555c27e4e6d86d1c31877279f6ac24bd349c4dc30b3eabdc7c8f99d4012c",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}

evaluation SHA-256: sha256:3f8bd7c23304478990523c0d754cd642ed46eb76db7c4784c47292f5803a7f61

Original data · JSON

{
  "args": [
    561
  ],
  "facts": [
    {
      "args": [
        "urn:case:flt:probatio",
        561,
        2
      ],
      "predicate": "propositum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560
      ],
      "predicate": "vestigium"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        1,
        2
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        2,
        4
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4,
        16
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        8,
        256
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        16,
        460
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        17,
        359
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        34,
        412
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        35,
        263
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        70,
        166
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        140,
        67
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        280,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        560,
        1
      ],
      "predicate": "residuum"
    },
    {
      "args": [
        "urn:case:flt:probatio",
        4
      ],
      "predicate": "divisor_allatus"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "la-gauss-theorema-fermatianum",
  "predicate": "primus",
  "proof": true
}

Итог: без предъявленного делителя критерий Ферма признаёт простым и составное число.

How to cite

The snapshot is immutable: the SHA-256 of the downloadable JSON pins it, so no access date is needed.

Citation

“Тест Ферма для 341 по основанию 2 и для 561 по основаниям 2, 5 и 7. Где критерий объявляет число простым и чем число Кармайкла отличается от обычного псевдопростого?”. Arxo Lens, as of 2026-09-06. https://lens.arxo.io/a/a_9qzwf1Xp4zT0njRr8U6CNa9t. Snapshot SHA-256: 47002a28deedbf9742ee6a67622cd4976d2b628aed2071e6157a9f17cce8d86c.

BibTeX

@misc{arxo-lens-a_9qzwf1Xp4zT0,
  title = {Тест Ферма для 341 по основанию 2 и для 561 по основаниям 2, 5 и 7. Где критерий объявляет число простым и чем число Кармайкла отличается от обычного псевдопростого?},
  howpublished = {Arxo Lens},
  url = {https://lens.arxo.io/a/a_9qzwf1Xp4zT0njRr8U6CNa9t},
  note = {as of 2026-09-06; SHA-256 47002a28deedbf9742ee6a67622cd4976d2b628aed2071e6157a9f17cce8d86c}
}

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