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Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients
T0 review · 0 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read Fermat-quotient power sums determine Wilson's theorem modulo every prime power.
desk verdict The paper proves a genuine general-n p-adic expansion of the Wilson quotient in terms of Fermat-quotient power sums; the proof is sound, and the only soft spots are minor exposition issues. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The mechanism is the pair (Bell polynomials, Newton's identities) acting on the product identity ∏(1+p q_p(a)) = (1-p W_p)^{p-1}. Expanding the left side gives elementary symmetric polynomials σ_ν(q_p), and Newton's identities rewrite σ_ν as polynomials in the power sums Q_p(1),...,Q_p(ν). Bell polynomials B_{n,k} supply the coefficients when powers of the unknown W_p are replaced by the assumed ψ-expansion; Lemma 3.3 extracts the p^ℓ coefficient from such powers. The output is the recursive family ψ_ν with ψ_1=x_1 and recurrence (4.1), which makes the congruences explicit and universal.
What would settle it
Directly compute both sides of Theorem 1.1(2) for a concrete pair with p>n, say p=11, n=5: evaluate W_11 modulo 11^5 as (10!+1)/11, compute Q_11(1),...,Q_11(5) from the Fermat quotients q_11(a), and compare with the displayed sum using the paper's computed ψ_5. Any mismatch modulo 11^5 falsifies the central claim.
Extended reading notes
Core claim
The central claim is Theorem 1.1. For every n≥1 there are unique multivariate polynomials ψ_ν(x_1,...,x_ν) ∈ Z[x_1,...,x_ν] with no constant term, computable recursively, such that for every odd prime p>n, W_p ≡ Σ_{ν=1}^{n} p^{ν-1}/ν! ψ_ν(Q_p(1),...,Q_p(ν)) mod p^n, and equivalently (p-1)! ≡ -1 + Σ_{ν=1}^{n} p^ν/ν! ψ_ν(Q_p(1),...,Q_p(ν)) mod p^{n+1}. Here Q_p(ν) is the ν-th power sum of Fermat quotients. The ψ_ν do not depend on p: once computed, they are universal coefficients for the p-adic development of the Wilson quotient at every order. The proof obtains them from a coefficient-extraction argument in which elementary symmetric polynomials of Fermat quotients are rewritten as power sums
Load-bearing premise
The argument depends on upgrading the coefficient congruence (3.10), which holds for all and infinitely many primes p>n, into an exact polynomial identity over Z that defines ψ_n; if this transfer is not valid, the ψ_n would not be universal and the recursive construction would break.
Editorial extensions
If this is right
- For any n and any odd prime p>n, the Wilson quotient modulo p^n can be computed from the n power sums Q_p(1),...,Q_p(n), with no separate prime-specific construction of the ψ_ν.
- The same sum, multiplied by p, gives (p-1)! modulo p^{n+1}, so Wilson's theorem is obtained at every higher prime power, not only modulo p.
- Because the ψ_ν are universal integer polynomials with an explicit recurrence, the computation can be automated; the bound #ψ_n ≤ PΣ(n) from Theorem 1.3 controls the number of terms.
- Corollary 1.2 gives a concrete evaluation scheme: at step ν, the power sum Q_p(ν) is only needed modulo p^{n-ν+1}.
- Theorem 4.3 makes the extreme terms of each ψ_n explicit: it starts with n! x_1 and ends with (-1)^{n-1}(n-1)! x_n.
Reading between the lines
- If the theorem holds, Wilson primes (W_p ≡ 0 mod p) and higher-order analogs could be probed by evaluating the universal sum rather than computing factorials digit by digit; the paper itself does not develop this application.
- The same coefficient-extraction mechanism may apply to other quotients of the form (a^m - 1)/m, not just Fermat quotients, since only the product-to-power-sum structure is used.
- The conjectured equality #ψ_n = PΣ(n) suggests that the recurrence is essentially minimal—no unexpected cancellations occur beyond those already visible; checking n=31 would test this directly.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves a p-adic refinement of Wilson's theorem. For n ≥ 1 and odd p > n, it establishes that the Wilson quotient W_p modulo p^n is a universal polynomial combination of the power sums Q_p(1), ..., Q_p(n) of Fermat quotients, and equivalently that (p−1)! modulo p^{n+1} satisfies a corresponding congruence. The polynomials ψν are shown to have integer coefficients and no constant term, and they are computed by an explicit recurrence (Theorem 4.1) involving Bell polynomials and signed Stirling numbers. The proof rests on an exact product identity (Lemma 3.1), Newton's identities, and iterative p-adic lifting, and it is supported by extensive tables and machine checks.
Significance. This is a genuine generalization of Lerch's classical congruence W_p ≡ Σ q_p(a) (mod p) to arbitrary p-adic order. The result is algorithmic and fully explicit, with no fitted or free parameters, and the companion paper [6] will extend it to Bernoulli numbers. The main proof is built from exact identities and standard lifting arguments rather than numerical speculation, and the conjectures are clearly labeled and separately evidenced. If correct, this is a solid contribution to the p-adic theory of Wilson and Fermat quotients and should be of interest to number theorists working on supercongruences and p-adic expansions of factorials.
minor comments (5)
- [§3, after (3.10)] The sentence 'Since congruence (3.10) holds for all and infinitely many p>n, so it also holds in Z such that ...' is misleading and, taken literally, not a valid inference: infinitely many prime moduli do not by themselves turn a congruence into an identity over Z. The existence of ψ_ℓ is better justified by explicitly defining it from the p-independent expression provided by Lemma 3.3, or by deferring to the exact recurrence in Theorem 4.1. This is a clarity issue rather than a substantive gap, but the sentence should be rewritten.
- [Theorem 1.1(1)] Uniqueness of the polynomials ψν is asserted but not separately proved. It follows from the unique recurrence in Theorem 4.1; the proof of Theorem 1.1 should say this explicitly.
- [§1 and §3] Minor typographical issues: 'A view years later' should be 'A few years later'; in (3.1) there is a doubled comma in bσ_ν(Q_p(1), ...,, Q_p(ν)).
- [Theorem 3.2] The notation W_{p,ℓ} is used in the statement before being defined. Please define W_{p,ℓ} as the p-adic approximation of W_p modulo p^ℓ in the statement of the theorem.
- [Corollary 4.4] The proof that #ψ_n ≤ P_Σ(n) is terse. It would help to state explicitly that every monomial of ψ_n has partition order at most n, so distinct monomials correspond to partitions with sum at most n, giving the bound P_Σ(n).
Circularity Check
No significant circularity: the derivation is self-contained, using exact product and Newton/Bell identities, with the polynomials psi_nu explicitly constructed.
full rationale
The paper's central claim is a congruence expansion of the Wilson quotient W_p in terms of power sums Q_p(nu) of Fermat quotients. The derivation chain is not circular. Lemma 3.1 starts from the exact product identity ∏_{a=1}^{p-1}(1+p q_p(a)) = (1 - p W_p)^{p-1}, which follows directly from the definitions q_p(a) = (a^{p-1}-1)/p and W_p = ((p-1)!+1)/p together with ∏ a = (p-1)!. This is an identity, not an assumption of the desired conclusion. The transition to power sums uses Newton's identities, a standard external algebraic fact. The iterative congruence in Theorem 3.2 is obtained by rewriting this exact expansion, not by importing the target congruence. In Theorem 3.4, the multivariate polynomials psi_nu are not fitted to data nor assumed from the conclusion: each psi_nu is explicitly obtained as the coefficient expression on the right-hand side after applying Lemma 3.3, which shows the coefficients are fixed integer polynomials independent of p. The sentence 'Since congruence (3.10) holds for all and infinitely many p>n, so it also holds in Z' is rhetorically odd, but it is not load-bearing; the coefficient extraction in Lemma 3.3 already defines psi_nu directly for every prime p>n. Theorem 4.1 then gives an explicit recurrence in terms of Bell polynomials and Stirling numbers, independent of W_p or Q_p. There are no fitted parameters, no post hoc selection of data, and no external benchmark whose construction encodes the result. The only self-citation is the forthcoming companion paper [6], which is mentioned for future Bernoulli-number translations and is not used in any proof. Thus the main theorem has independent mathematical content and no circular dependence on its own conclusion.
Assumptions & free parameters
assumptions (4)
- standard math Generating function of partial Bell polynomials (2.2) and Newton identities (2.4) give integral expressions for elementary symmetric polynomials in terms of power sums.
- standard math Fermat's little theorem and Wilson's theorem hold, so q_p(a) and W_p are integers for odd primes p.
- standard math p-adic congruence arithmetic: terms substituted at the correct valuation give valid congruences modulo p^ell.
- domain assumption Congruence (3.10) holding modulo p for infinitely many primes p>n forces a universal polynomial identity over Z.
Cite this review
Pith. "Pith review of Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients." pith.science (2026). https://pith.science/paper/HAPIDW56
@misc{pith2026250905235,
author = {Pith},
title = {Pith review of: Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients},
year = {2026},
howpublished = {\url{https://pith.science/paper/HAPIDW56}},
note = {Machine review of arXiv:2509.05235}
}
read the original abstract
We show that Wilson's theorem as well as the Wilson quotient can be described by supercongruences modulo any higher prime power involving terms of power sums of Fermat quotients. The new approach uses Bell polynomials and Newton's identities relating elementary symmetric polynomials to power sums. This enables us to compute certain multivariate polynomials recursively that are needed to establish the supercongruences. Subsequently, we give a recurrence formula for these polynomials and show further properties.
Forward citations
Cited by 1 Pith paper
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Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products
Odd-order reciprocal power sums satisfy a uniform Bernoulli-polynomial congruence modulo n, and Lehmer-type products admit truncated Bell-polynomial expansions modulo n^{K+1}.
Reference graph
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Reviewed August 5, 2026 · model on record in the stance chip above.
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