{"assistant":{"explanations":[{"calculationRefs":["presumed-7"],"text":"Опорный случай: для настоящего простого числа критерий даёт верный ответ."},{"calculationRefs":["pseudo-341"],"text":"След степени 340 по модулю 341 сходится к единице: тест пройден, хотя 341 = 11 · 31."},{"calculationRefs":["primus-341"],"text":"Делитель не предъявлен, критерий пройден — вывод «простое» следует из посылок, и в этом слабость теста."},{"calculationRefs":["carmichael-2"],"text":"Первое основание тест не различает."},{"calculationRefs":["carmichael-5"],"text":"Второе основание тоже проходит: у обычного псевдопростого здесь была бы осечка."},{"calculationRefs":["carmichael-7"],"text":"И третье. Число Кармайкла обманывает все взаимно простые основания сразу."},{"calculationRefs":["primus-561"],"text":"Итог: без предъявленного делителя критерий Ферма признаёт простым и составное число."}],"origin":"assistant","summary":"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. 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основание, по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#CatenaInitium","kind":"rule","labels":[{"language":"la","status":"official","text":"In omni progressione geometrica \\(1, a, aa, a^3\\) etc. praeter primum 1"},{"language":"ru","status":"unofficial","text":"начало цепи: первое звено следа обязано быть остатком самого основания по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#Probatio","kind":"type_decl","labels":[{"language":"la","status":"official","text":"probatio"},{"language":"ru","status":"unofficial","text":"испытание — модуль, основание и предъявленный след возведения, о которых задан вопрос дела"}],"name":"Probatio","package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta","kind":"rule","labels":[{"language":"la","status":"official","text":"sive \\(a^{(p-1)}-1\\) semper per \\(p\\) divisibilis est, quando p est primus ipsum a non metiens"},{"language":"ru","status":"unofficial","text":"теорема Ферма как тест: проверенная цепь дошла до показателя на единицу меньше модуля, остаток там единица, и модуль основания не делит"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#TheoremaFermatianumSemperTenet","kind":"constraint","labels":[{"language":"la","status":"unofficial","text":"theorema Fermatianum nullo casu violatur"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: экземпляр теоремы Ферма не нарушается ни одним основанием — при модуле 3, 5 либо 7 основание, модулем не делящееся, обязано давать единицу в степени на единицу меньше модуля"}],"package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"casus propositus"},{"language":"ru","status":"unofficial","text":"антецедент инварианта: испытанию предъявлены модуль и основание"}],"name":"casus_propositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"catena potestatum ad exponentem usque probata"},{"language":"ru","status":"unofficial","text":"след проверен непрерывно от показателя 1 до этого показателя — каждое звено сошлось по правилу шага (артикул 46)"}],"name":"catena","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus compositus: divisorem habet"},{"language":"ru","status":"unofficial","text":"модуль составной — у него предъявлен и проверен собственный делитель; иных оснований составности пакет не выводит (разбор отвергнутого обращения артикула 49 — в audit §4)"}],"name":"compositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#primus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus primus"},{"language":"ru","status":"unofficial","text":"модуль презюмируется простым — тест Ферма пройден и составность не установлена; шаг ДОКТРИНАЛЬНЫЙ, обратной теоремы у Гаусса нет"}],"name":"primus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#primus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"probatio Fermatiana transacta: potestas p-1 unitati congrua"},{"language":"ru","status":"unofficial","text":"тест Ферма пройден — проверенная цепь дошла до показателя на единицу меньше модуля, и остаток там единица (артикул 50)"}],"name":"probatio_fermatiana_transacta","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"progressio geometrica secundum modulum p ad basim a"},{"language":"ru","status":"unofficial","text":"модуль и основание геометрической прогрессии, предъявленные испытанию (артикул 45: «In omni progressione geometrica … secundum modulum p ad a primum»)"}],"name":"propositum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/basis","labels":[],"name":"basis","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"residuum minimum potestatis exponentis dati"},{"language":"ru","status":"unofficial","text":"звено следа: дело заявляет, что основание в степени exponens сравнимо с valor по модулю испытания; движок звено не вычисляет, а проверяет"}],"name":"residuum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/valor","labels":[],"name":"valor","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"vestigium potestatum ab exponente uno usque ad longitudinem datam"},{"language":"ru","status":"unofficial","text":"дело объявляет, что след предъявлен непрерывно от показателя 1 до этой длины; сверку объявления с самим следом ведёт сторож vestigium_ruptum"}],"name":"vestigium","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/longitudo","labels":[],"name":"longitudo","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"violatio theorematis Fermatiani"},{"language":"ru","status":"unofficial","text":"носитель нарушения теоремы Ферма: у основания, модулем не делящегося, степень на единицу меньше модуля не дала единицы"}],"name":"violatio_fermatiana","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"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:c0ebafc9508a878bc4720fafa551848fbb32b28b4fa7ce5f72a869c4a50c37f1","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810","resultHash":"sha256:3e03779ab824b4c6189b1554082b9050407f34c22da7da5a54c67d83a4ee5be7","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"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},"sources":[{"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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"primus: **TRUE_ONLY** — установлено\nВыведено правом: 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)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(7)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — 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основание, по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#CatenaInitium","kind":"rule","labels":[{"language":"la","status":"official","text":"In omni progressione geometrica \\(1, a, aa, a^3\\) etc. praeter primum 1"},{"language":"ru","status":"unofficial","text":"начало цепи: первое звено следа обязано быть остатком самого основания по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#Probatio","kind":"type_decl","labels":[{"language":"la","status":"official","text":"probatio"},{"language":"ru","status":"unofficial","text":"испытание — модуль, основание и предъявленный след возведения, о которых задан вопрос дела"}],"name":"Probatio","package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta","kind":"rule","labels":[{"language":"la","status":"official","text":"sive \\(a^{(p-1)}-1\\) semper per \\(p\\) divisibilis est, quando p est primus ipsum a non metiens"},{"language":"ru","status":"unofficial","text":"теорема Ферма как тест: проверенная цепь дошла до показателя на единицу меньше модуля, остаток там единица, и модуль основания не делит"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#TheoremaFermatianumSemperTenet","kind":"constraint","labels":[{"language":"la","status":"unofficial","text":"theorema Fermatianum nullo casu violatur"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: экземпляр теоремы Ферма не нарушается ни одним основанием — при модуле 3, 5 либо 7 основание, модулем не делящееся, обязано давать единицу в степени на единицу меньше модуля"}],"package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"casus propositus"},{"language":"ru","status":"unofficial","text":"антецедент инварианта: испытанию предъявлены модуль и основание"}],"name":"casus_propositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"catena potestatum ad exponentem usque probata"},{"language":"ru","status":"unofficial","text":"след проверен непрерывно от показателя 1 до этого показателя — каждое звено сошлось по правилу шага (артикул 46)"}],"name":"catena","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus compositus: divisorem habet"},{"language":"ru","status":"unofficial","text":"модуль составной — у него предъявлен и проверен собственный делитель; иных оснований составности пакет не выводит (разбор отвергнутого обращения артикула 49 — в audit §4)"}],"name":"compositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#primus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus primus"},{"language":"ru","status":"unofficial","text":"модуль презюмируется простым — тест Ферма пройден и составность не установлена; шаг ДОКТРИНАЛЬНЫЙ, обратной теоремы у Гаусса нет"}],"name":"primus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#primus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"probatio Fermatiana transacta: potestas p-1 unitati congrua"},{"language":"ru","status":"unofficial","text":"тест Ферма пройден — проверенная цепь дошла до показателя на единицу меньше модуля, и остаток там единица (артикул 50)"}],"name":"probatio_fermatiana_transacta","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"progressio geometrica secundum modulum p ad basim a"},{"language":"ru","status":"unofficial","text":"модуль и основание геометрической прогрессии, предъявленные испытанию (артикул 45: «In omni progressione geometrica … secundum modulum p ad a primum»)"}],"name":"propositum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/basis","labels":[],"name":"basis","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"residuum minimum potestatis exponentis dati"},{"language":"ru","status":"unofficial","text":"звено следа: дело заявляет, что основание в степени exponens сравнимо с valor по модулю испытания; движок звено не вычисляет, а проверяет"}],"name":"residuum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/valor","labels":[],"name":"valor","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"vestigium potestatum ab exponente uno usque ad longitudinem datam"},{"language":"ru","status":"unofficial","text":"дело объявляет, что след предъявлен непрерывно от показателя 1 до этой длины; сверку объявления с самим следом ведёт сторож vestigium_ruptum"}],"name":"vestigium","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/longitudo","labels":[],"name":"longitudo","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"violatio theorematis Fermatiani"},{"language":"ru","status":"unofficial","text":"носитель нарушения теоремы Ферма: у основания, модулем не делящегося, степень на единицу меньше модуля не дала единицы"}],"name":"violatio_fermatiana","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"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"},"request":{"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},"sources":[{"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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"probatio_fermatiana_transacta: **TRUE_ONLY** — установлено\nВыведено 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)\n…и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(341)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…)\nproof-граф: 32 узлов — поле evaluation готово для 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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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"primus: **TRUE_ONLY** — установлено\nВыведено правом: primus(341)\n…и ещё 14 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(341)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…)\nproof-граф: 32 узлов — поле evaluation готово для 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основание, по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#CatenaInitium","kind":"rule","labels":[{"language":"la","status":"official","text":"In omni progressione geometrica \\(1, a, aa, a^3\\) etc. praeter primum 1"},{"language":"ru","status":"unofficial","text":"начало цепи: первое звено следа обязано быть остатком самого основания по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#Probatio","kind":"type_decl","labels":[{"language":"la","status":"official","text":"probatio"},{"language":"ru","status":"unofficial","text":"испытание — модуль, основание и предъявленный след возведения, о которых задан вопрос дела"}],"name":"Probatio","package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta","kind":"rule","labels":[{"language":"la","status":"official","text":"sive \\(a^{(p-1)}-1\\) semper per \\(p\\) divisibilis est, quando p est primus ipsum a non metiens"},{"language":"ru","status":"unofficial","text":"теорема Ферма как тест: проверенная цепь дошла до показателя на единицу меньше модуля, остаток там единица, и модуль основания не делит"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#TheoremaFermatianumSemperTenet","kind":"constraint","labels":[{"language":"la","status":"unofficial","text":"theorema Fermatianum nullo casu violatur"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: экземпляр теоремы Ферма не нарушается ни одним основанием — при модуле 3, 5 либо 7 основание, модулем не делящееся, обязано давать единицу в степени на единицу меньше модуля"}],"package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"casus propositus"},{"language":"ru","status":"unofficial","text":"антецедент инварианта: испытанию предъявлены модуль и основание"}],"name":"casus_propositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"catena potestatum ad exponentem usque probata"},{"language":"ru","status":"unofficial","text":"след проверен непрерывно от показателя 1 до этого показателя — каждое звено сошлось по правилу шага (артикул 46)"}],"name":"catena","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus compositus: divisorem habet"},{"language":"ru","status":"unofficial","text":"модуль составной — у него предъявлен и проверен собственный делитель; иных оснований составности пакет не выводит (разбор отвергнутого обращения артикула 49 — в audit §4)"}],"name":"compositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#primus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus primus"},{"language":"ru","status":"unofficial","text":"модуль презюмируется простым — тест Ферма пройден и составность не установлена; шаг ДОКТРИНАЛЬНЫЙ, обратной теоремы у Гаусса нет"}],"name":"primus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#primus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"probatio Fermatiana transacta: potestas p-1 unitati congrua"},{"language":"ru","status":"unofficial","text":"тест Ферма пройден — проверенная цепь дошла до показателя на единицу меньше модуля, и остаток там единица (артикул 50)"}],"name":"probatio_fermatiana_transacta","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"progressio geometrica secundum modulum p ad basim a"},{"language":"ru","status":"unofficial","text":"модуль и основание геометрической прогрессии, предъявленные испытанию (артикул 45: «In omni progressione geometrica … secundum modulum p ad a primum»)"}],"name":"propositum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/basis","labels":[],"name":"basis","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"residuum minimum potestatis exponentis dati"},{"language":"ru","status":"unofficial","text":"звено следа: дело заявляет, что основание в степени exponens сравнимо с valor по модулю испытания; движок звено не вычисляет, а проверяет"}],"name":"residuum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/valor","labels":[],"name":"valor","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"vestigium potestatum ab exponente uno usque ad longitudinem datam"},{"language":"ru","status":"unofficial","text":"дело объявляет, что след предъявлен непрерывно от показателя 1 до этой длины; сверку объявления с самим следом ведёт сторож vestigium_ruptum"}],"name":"vestigium","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/longitudo","labels":[],"name":"longitudo","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"violatio theorematis Fermatiani"},{"language":"ru","status":"unofficial","text":"носитель нарушения теоремы Ферма: у основания, модулем не делящегося, степень на единицу меньше модуля не дала единицы"}],"name":"violatio_fermatiana","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"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"},"request":{"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},"sources":[{"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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"probatio_fermatiana_transacta: **TRUE_ONLY** — установлено\nВыведено 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)\n…и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…)\nproof-граф: 32 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["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, 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основание, по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#CatenaInitium","kind":"rule","labels":[{"language":"la","status":"official","text":"In omni progressione geometrica \\(1, a, aa, a^3\\) etc. praeter primum 1"},{"language":"ru","status":"unofficial","text":"начало цепи: первое звено следа обязано быть остатком самого основания по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#Probatio","kind":"type_decl","labels":[{"language":"la","status":"official","text":"probatio"},{"language":"ru","status":"unofficial","text":"испытание — модуль, основание и предъявленный след возведения, о которых задан вопрос дела"}],"name":"Probatio","package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta","kind":"rule","labels":[{"language":"la","status":"official","text":"sive \\(a^{(p-1)}-1\\) semper per \\(p\\) divisibilis est, quando p est primus ipsum a non metiens"},{"language":"ru","status":"unofficial","text":"теорема Ферма как тест: проверенная цепь дошла до показателя на единицу меньше модуля, остаток там единица, и модуль основания не делит"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#TheoremaFermatianumSemperTenet","kind":"constraint","labels":[{"language":"la","status":"unofficial","text":"theorema Fermatianum nullo casu violatur"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: экземпляр теоремы Ферма не нарушается ни одним основанием — при модуле 3, 5 либо 7 основание, модулем не делящееся, обязано давать единицу в степени на единицу меньше модуля"}],"package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"casus propositus"},{"language":"ru","status":"unofficial","text":"антецедент инварианта: испытанию предъявлены модуль и основание"}],"name":"casus_propositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"catena potestatum ad exponentem usque probata"},{"language":"ru","status":"unofficial","text":"след проверен непрерывно от показателя 1 до этого показателя — каждое звено сошлось по правилу шага (артикул 46)"}],"name":"catena","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus compositus: divisorem habet"},{"language":"ru","status":"unofficial","text":"модуль составной — у него предъявлен и проверен собственный делитель; иных оснований составности пакет не выводит (разбор отвергнутого обращения артикула 49 — в audit §4)"}],"name":"compositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#primus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus primus"},{"language":"ru","status":"unofficial","text":"модуль презюмируется простым — тест Ферма пройден и составность не установлена; шаг ДОКТРИНАЛЬНЫЙ, обратной теоремы у Гаусса нет"}],"name":"primus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#primus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"probatio Fermatiana transacta: potestas p-1 unitati congrua"},{"language":"ru","status":"unofficial","text":"тест Ферма пройден — проверенная цепь дошла до показателя на единицу меньше модуля, и остаток там единица (артикул 50)"}],"name":"probatio_fermatiana_transacta","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"progressio geometrica secundum modulum p ad basim a"},{"language":"ru","status":"unofficial","text":"модуль и основание геометрической прогрессии, предъявленные испытанию (артикул 45: «In omni progressione geometrica … secundum modulum p ad a primum»)"}],"name":"propositum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/basis","labels":[],"name":"basis","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"residuum minimum potestatis exponentis dati"},{"language":"ru","status":"unofficial","text":"звено следа: дело заявляет, что основание в степени exponens сравнимо с valor по модулю испытания; движок звено не вычисляет, а проверяет"}],"name":"residuum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/valor","labels":[],"name":"valor","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"vestigium potestatum ab exponente uno usque ad longitudinem datam"},{"language":"ru","status":"unofficial","text":"дело объявляет, что след предъявлен непрерывно от показателя 1 до этой длины; сверку объявления с самим следом ведёт сторож vestigium_ruptum"}],"name":"vestigium","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/longitudo","labels":[],"name":"longitudo","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"violatio theorematis Fermatiani"},{"language":"ru","status":"unofficial","text":"носитель нарушения теоремы Ферма: у основания, модулем не делящегося, степень на единицу меньше модуля не дала единицы"}],"name":"violatio_fermatiana","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"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:dee415fe1d3029ea6b9fd6301e36b28728464b6bcedc24ea80c9868603bccf14","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810","resultHash":"sha256:899a16675679ae3bc6a25bedba834acf9fe5cfbacb9ecec2c96704e7bcb88173","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"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},"sources":[{"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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"probatio_fermatiana_transacta: **TRUE_ONLY** — установлено\nВыведено 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)\n…и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…)\nproof-граф: 32 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["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, 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основание, по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#CatenaInitium","kind":"rule","labels":[{"language":"la","status":"official","text":"In omni progressione geometrica \\(1, a, aa, a^3\\) etc. praeter primum 1"},{"language":"ru","status":"unofficial","text":"начало цепи: первое звено следа обязано быть остатком самого основания по модулю испытания"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R1","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#PrimusPraesumptus/R2","kind":"rule","labels":[{"language":"la","status":"unofficial","text":"numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur"},{"language":"ru","status":"unofficial","text":"доктринальный шаг: модуль, прошедший тест Ферма, презюмируется простым, пока составность не установлена; у Гаусса обратной теоремы нет, и строгим этот вывод был бы враньём о числе 341"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"defeasible"},{"id":"urn:la:gauss:clir:theorema-fermatianum#Probatio","kind":"type_decl","labels":[{"language":"la","status":"official","text":"probatio"},{"language":"ru","status":"unofficial","text":"испытание — модуль, основание и предъявленный след возведения, о которых задан вопрос дела"}],"name":"Probatio","package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#ProbatioFermatianaTransacta","kind":"rule","labels":[{"language":"la","status":"official","text":"sive \\(a^{(p-1)}-1\\) semper per \\(p\\) divisibilis est, quando p est primus ipsum a non metiens"},{"language":"ru","status":"unofficial","text":"теорема Ферма как тест: проверенная цепь дошла до показателя на единицу меньше модуля, остаток там единица, и модуль основания не делит"}],"package":"urn:la:gauss:clir:theorema-fermatianum","strength":"strict"},{"id":"urn:la:gauss:clir:theorema-fermatianum#TheoremaFermatianumSemperTenet","kind":"constraint","labels":[{"language":"la","status":"unofficial","text":"theorema Fermatianum nullo casu violatur"},{"language":"ru","status":"unofficial","text":"инвариант §93.1: экземпляр теоремы Ферма не нарушается ни одним основанием — при модуле 3, 5 либо 7 основание, модулем не делящееся, обязано давать единицу в степени на единицу меньше модуля"}],"package":"urn:la:gauss:clir:theorema-fermatianum"},{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"casus propositus"},{"language":"ru","status":"unofficial","text":"антецедент инварианта: испытанию предъявлены модуль и основание"}],"name":"casus_propositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#casus_propositus/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"catena potestatum ad exponentem usque probata"},{"language":"ru","status":"unofficial","text":"след проверен непрерывно от показателя 1 до этого показателя — каждое звено сошлось по правилу шага (артикул 46)"}],"name":"catena","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#catena/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus compositus: divisorem habet"},{"language":"ru","status":"unofficial","text":"модуль составной — у него предъявлен и проверен собственный делитель; иных оснований составности пакет не выводит (разбор отвергнутого обращения артикула 49 — в audit §4)"}],"name":"compositus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#compositus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#primus","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"numerus primus"},{"language":"ru","status":"unofficial","text":"модуль презюмируется простым — тест Ферма пройден и составность не установлена; шаг ДОКТРИНАЛЬНЫЙ, обратной теоремы у Гаусса нет"}],"name":"primus","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#primus/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"probatio Fermatiana transacta: potestas p-1 unitati congrua"},{"language":"ru","status":"unofficial","text":"тест Ферма пройден — проверенная цепь дошла до показателя на единицу меньше модуля, и остаток там единица (артикул 50)"}],"name":"probatio_fermatiana_transacta","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#probatio_fermatiana_transacta/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"progressio geometrica secundum modulum p ad basim a"},{"language":"ru","status":"unofficial","text":"модуль и основание геометрической прогрессии, предъявленные испытанию (артикул 45: «In omni progressione geometrica … secundum modulum p ad a primum»)"}],"name":"propositum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/modulus","labels":[],"name":"modulus","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#propositum/arg/basis","labels":[],"name":"basis","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"residuum minimum potestatis exponentis dati"},{"language":"ru","status":"unofficial","text":"звено следа: дело заявляет, что основание в степени exponens сравнимо с valor по модулю испытания; движок звено не вычисляет, а проверяет"}],"name":"residuum","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/exponens","labels":[],"name":"exponens","type":{"name":"urn:law:std#Integer"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#residuum/arg/valor","labels":[],"name":"valor","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"vestigium potestatum ab exponente uno usque ad longitudinem datam"},{"language":"ru","status":"unofficial","text":"дело объявляет, что след предъявлен непрерывно от показателя 1 до этой длины; сверку объявления с самим следом ведёт сторож vestigium_ruptum"}],"name":"vestigium","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}},{"id":"urn:la:gauss:clir:theorema-fermatianum#vestigium/arg/longitudo","labels":[],"name":"longitudo","type":{"name":"urn:law:std#Integer"}}],"symbolKind":"relation"},{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana","kind":"symbol_decl","labels":[{"language":"la","status":"unofficial","text":"violatio theorematis Fermatiani"},{"language":"ru","status":"unofficial","text":"носитель нарушения теоремы Ферма: у основания, модулем не делящегося, степень на единицу меньше модуля не дала единицы"}],"name":"violatio_fermatiana","package":"urn:la:gauss:clir:theorema-fermatianum","parameters":[{"id":"urn:la:gauss:clir:theorema-fermatianum#violatio_fermatiana/arg/pr","labels":[],"name":"pr","type":{"name":"urn:la:gauss:clir:theorema-fermatianum#Probatio"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"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:c3648e6e8d1a57db7dd88a960e2293b4d68d864fef04bae21cf03f680ddbea76","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:d76c76b2bf97f9e2baf8dccf465e15d3843a8f2b202f96b0bee8eca759231810","resultHash":"sha256:6cc12d7752756b52b7742e3cf14970f03e79579644fdd808e7f325a4800bf158","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"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},"sources":[{"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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"probatio_fermatiana_transacta: **TRUE_ONLY** — установлено\nВыведено 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)\n…и ещё 1 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…)\nproof-граф: 32 узлов — поле evaluation готово для 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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 $\\textstyle a^{r} \\equiv a^{\\rho} \\pmod{t}$."}]},{"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."}]}],"text":"primus: **TRUE_ONLY** — установлено\nВыведено правом: primus(561)\n…и ещё 14 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: CasusPropositus, CatenaDuplicata, CatenaGradus, CatenaInitium, PrimusPraesumptus/R1, ProbatioFermatianaTransacta\nОтвет поражаем правилом «numerus, cuius probatio Fermatiana transacta est, primus praesumitur donec compositus probetur» — оно отменило бы вывод, будь установлено: compositus(561)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Малая теорема Ферма: Disquisitiones arithmeticae Гаусса, артикулы 45—50 «De residuis potestatum» — доктрина — EXECUTABLE 6 §33.1 (programHash sha256:d76c76b2bf97…)\nproof-граф: 32 узлов — поле evaluation готово для law_explain"}],"language":"ru","question":{"origin":"user","text":"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?"},"schemaVersion":"law.answers.document/0.1"}