Une supercongruence de Wilson exacte pour les premiers d'Artin-Schreier de F_p[t] (An exact Wilson supercongruence for Artin-Schreier primes in F_p[t])
Bibliographic record
Abstract
Soit p un nombre premier impair et wp = t^p - t - m (m dans F_p^*) un premierd'Artin-Schreier de F_p[t] ; ce sont exactement les premiers de Wilson de degrep au sens de Thakur. En notant F_p le produit de tous les polynomes non nuls dedegre inferieur a p (de sorte que le theoreme de Wilson en corps de fonctionsdonne F_p == -1 mod wp), on demontre la supercongruence EXACTE F_p == -1 - wp^(p-1) (mod wp^p). Par consequent la congruence de Wilson tient exactement a l'ordre p-1 (et non p),avec terme dominant explicite (F_p + 1)/wp^(p-1) == -1 (mod wp). Thakur (2012,2013, 2015, 2022) etablit seulement la borne inferieure (mod wp^(p-1)) et posel'exactitude de l'ordre et le calcul des multiplicites comme question ouverte(Thakur 2022, Question 7.5(1)) pour p > 2. Le present resultat repond a cettequestion pour la famille canonique des premiers d'Artin-Schreier : il n'y a pasd'amelioration au-dela de wp^(p-1), et l'obstruction est exactement -wp^(p-1).La demonstration repose sur le developpement wp-adique exact des crochets[i] = t^(p^i) - t (a savoir [i] == i*m + wp mod wp^p), l'identite L_(p-1) ==wp^(p-1) - 1 mod wp^p, et une factorisation de F_p comme -(produit des polynomesunitaires de degre < p)^(p-1). Un script Python pur (compatible Pyodide, joint)verifie le theoreme en arithmetique exacte pour p = 3, 5, 7, 11, en calculant enoutre F_p DIRECTEMENT (produit de tous les polynomes non nuls de degre < p) pourp = 3 et p = 5. [EN] Let p be an odd prime and wp = t^p - t - m (m in F_p^*) an Artin-Schreierprime of F_p[t] -- exactly the degree-p Wilson primes of Thakur. With F_p theproduct of all nonzero polynomials of degree < p (so that the function-fieldWilson theorem gives F_p == -1 mod wp), we prove the EXACT supercongruenceF_p == -1 - wp^(p-1) (mod wp^p). Hence the Wilson congruence holds to orderexactly p-1 (not p), with explicit leading term (F_p+1)/wp^(p-1) == -1 (mod wp).Thakur (2012-2022) proves only the lower bound (mod wp^(p-1)) and lists exactnessof the order and the exact multiplicities as an open question for p > 2; thisanswers it for the canonical Artin-Schreier family. An enclosed pure-Python(Pyodide-compatible) script verifies the theorem in exact arithmetic forp = 3, 5, 7, 11, computing F_p directly for p = 3, 5.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".