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The exact multiplicity of the Wilson supercongruence in F_q[t]

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Let q = p^s with p an odd prime, and let P be a monic irreducible polynomial of degree d >= 2 in F_q[t]. The Wilson analogue F_d = -1 mod P holds for the product F_d of all nonzero polynomials of degree less than d, and Thakur proved that P is a Wilson prime, i.e. F_d = -1 mod P^2, exactly when all exponents of P are congruent to 0 or 1 mod p, in which case the congruence holds modulo P^(p-1). He asked whether it can hold modulo a higher power, and what the exact multiplicities are. Let r be the largest integer such that all exponents of P are congruent to 0 or 1 modulo p^r. We prove that the exact multiplicity m = v_P(F_d + 1) is 1 if r = 0, p^r - 1 if 1 <= r <= s, and q + 1 if r > s. In particular m <= q + 1, and the multiplicity of a Wilson prime in the numerator of the sum over i < d of 1/(t^(q^i) - t) is exactly m - 1. Examples over F_3, F_5 and F_9 show that all regimes occur. A script checks the theorem on all primes of small degree and on these examples.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by an independent AI review and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. J. Sauerberg, L. Shu, D. S. Thakur and G. Todd, Infinitude of Wilson primes for $\Fq[t]$, Acta Arith. 157 (2013), 91--100.
  2. D. S. Thakur, Differential characterization of Wilson primes for $\Fq[t]$, Algebra Number Theory 7 (2013), 1841--1848.
  3. D. S. Thakur, Fermat--Wilson supercongruences, arithmetic derivatives and strange factorizations, J. Théor. Nombres Bordeaux 36 (2024), no. 2, 481--491, doi:10.5802/jtnb.1285; arXiv:2211.01076.arXivDOI

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026