{"id":"4aa710ea-793b-4294-93a8-f3f55d6011c5","arxiv_id":"2509.05235","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every n and odd prime p>n, the Wilson quotient modulo p^n is a finite sum of recursively defined integer polynomials evaluated at power sums of Fermat quotients.","lead":"This paper proves a higher prime power version of Wilson's theorem, expressing the Wilson quotient modulo p^n through power sums of Fermat quotients using recursively defined integer polynomials. A generalist should care because it replaces scattered low-power cases with one systematic framework that connects to Bernoulli numbers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's weakest_assumption pointed to the transfer from congruence (3.10) to a polynomial identity. My analysis shows that this step is not actually load-bearing because the polynomial ψ_ℓ is explicitly defined by the right-hand side of (3.10), whose coefficients are independent of p and integral by Lemma 3.3. The phrase about infinitely many primes is unnecessary and perhaps misleading, but it does not conceal a gap. I verified the recurrence (4.2) for small n and found it consistent with the tables when signed Stirling numbers are used. The paper's central claim is the congruence for W_p; the proof provides a constructive induction. No concrete flaw emerged. Thus the verdict remains ACCEPT (UNCHANGED), and the concern flagged by the reader is more a matter of exposition than of mathematical soundness.","tokens_in":10930,"tokens_out":56586,"duration_ms":463233,"concrete_test":"As a verification step, symbolically expand the recurrence (4.1) for n=4 using the definition of Ψ_4 in (4.2) and compare the resulting polynomial with ψ_4 from Table A.4. Additionally, compute W_11 mod 11^4 using the theorem's formula and compare with the direct value ((10)!+1)/11 mod 11^4; this tests the congruence for a prime p>n with n=4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After scrutinizing the proof, the allegedly weakest step—the inference of an exact polynomial identity from congruence (3.10) for infinitely many p—does not actually bear weight. In Theorem 3.4, the polynomial ψ_ℓ is explicitly constructed as the right-hand side of (3.10): it is a fixed polynomial in x_1,...,x_ℓ because the coefficients [p^{ℓ-1-ν}] S_{p,ℓ,ν} are extracted from expressions that depend on p only through polynomials like (p-1)_{ν+1}, whose coefficients are Stirling numbers independent of p. Lemma 3.3 ensures that after multiplying by ℓ!, the resulting expression has integer coefficients. Thus ψ_ℓ is defined directly, and the congruence T_{p,ℓ} ≡ ψ_ℓ(Q_p) (mod p) holds by construction for every prime p>n; no infinite-prime transfer is needed. The verbose phrase 'holds for all and infinitely many p>n' appears to be a rhetorical artefact rather than a logical necessity. I also checked the recurrence (4.2) for small n; the sign of the Stirling numbers must be interpreted as signed (as defined in the paper), and with that the formula reproduces the provided tables. No internal inconsistency found. The main theorem appears correct, and the proof, though intricate, is structurally sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11239,"tokens_out":22588,"duration_ms":202406,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§3, after (3.10)"},{"comment":"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.","section":"Theorem 1.1(1)"},{"comment":"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)).","section":"§1 and §3"},{"comment":"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.","section":"Theorem 3.2"},{"comment":"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).","section":"Corollary 4.4"}],"recommendation":"minor_revision","confidential_remarks":"The paper is within the scope of the journal and the central result appears sound. The only point that might trouble a careful reader is the 'infinitely many primes' sentence in §3, but it is repairable and does not affect the correctness of Theorem 4.1. I see no grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result, worth refereeing. It generalizes Lerch's mod p congruence and the Glaisher/Sun low-power cases to a uniform expansion of W_p modulo p^n for every n, with explicit integer polynomials ψ_ν computed recursively. The method is clean: start with the exact product identity (1 + p q_p(a)) product over a = (1 - p W_p)^{p-1}, expand, then use Newton identities and Bell polynomials to eliminate the W_p terms iteratively. The main theorem 1.1 gives the congruence and the factorial equivalent. Theorem 4.1 gives an explicit recurrence for the ψ_ν involving Stirling and Bell polynomials. That is new, not a repackaging of Lerch. The bound #ψ_n ≤ P_Σ(n) is a nice extra.\n\nWhere are the soft spots? The uniqueness clause in Theorem 1.1(1) is asserted rather than proved, but that is minor; the recurrence defines ψ_ν explicitly, so uniqueness follows. The phrase in Theorem 3.4 about 'holds for all and infinitely many p' reads as an infinite-prime transfer, but as the stress-test note says, it actually does nothing: ψ_ℓ is defined constructively as the right-hand side of (3.10), with coefficients extracted from expressions that are p-independent. So that suspicious sentence is just clumsy writing, not a logical gap. The conjectures (1.4 and 4.6) are clearly labeled; they don't affect the main theorem. No code is shipped, but the recurrence is explicit enough to reproduce the tables by hand, and the first terms check out. The only self-citation is the forthcoming companion paper, which is not used in the proof.\n\nBottom line: a solid, self-contained proof of a genuine generalization. The paper is written for number theorists working on congruences and Fermat/Wilson quotients; it will also be useful for the announced Bernoulli-number translation. I would send it to a serious referee.","headline":"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.","tokens_in":11663,"tokens_out":1788,"would_cite":true,"duration_ms":19894,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B65","11A07","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fermat-quotient power sums determine Wilson's theorem modulo every prime power.","keywords":["Fermat quotient","Wilson quotient","Wilson's theorem","Bell polynomials","symmetric polynomials","supercongruence","power sums","p-adic expansion"],"falsifier":"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.","tokens_in":10872,"feed_emoji":"🔢","tokens_out":5801,"duration_ms":59849,"temperature":0.7,"pith_summary":"This paper aims to establish that Wilson's congruence and the Wilson quotient are not just one-modulus facts: modulo any prime power p^n, they are fully described by the power sums of Fermat quotients. To do this, the paper introduces a family of integer polynomials ψ_ν, defined independently of p, that can be computed recursively and that supply every term of the p-adic expansion. If the main theorem is right, Lerch's 1905 congruence W_p ≡ Σ q_p(a) mod p becomes the first step of a complete ladder, with no unknown constants at any order. The payoff is that a multiplicative problem—the factorial modulo p^n—is converted into additive power-sum data that can be evaluated algorithmically.","feed_headline":"Fermat-quotient sums give Wilson's quotient modulo any p^n","feed_subtitle":"Universal recursively built polynomials turn the factorial congruence into additive power-sum data.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the base congruence W_p ≡ Σ q_p(a) mod p (Lerch), which is the ℓ=1 first step of the ladder.","marker":"[8]"},{"why":"Supplies the polynomial relation whose product form generates the key identity of Lemma 3.1.","marker":"[7]"},{"why":"Defines the Bell polynomials used to extract coefficients and express symmetric polynomials in terms of power sums.","marker":"[2]"},{"why":"Provides the Newton-identity and Bell-polynomial machinery, including the form of the σ^★_k polynomials used in the recurrence.","marker":"[3]"}],"fun_headline_variants":["From Fermat power sums: Wilson quotient at all p^n","Universal polynomials give Wilson's quotient mod p^n","Bell-built polynomials crack Wilson's quotient at any p^n","Wilson quotient via Fermat sums: one polynomial fits all p^n","Fermat sums unlock Wilson's quotient modulo any p^n"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["From Fermat power sums: Wilson quotient at all p^n","Universal polynomials give Wilson's quotient mod p^n","Bell-built polynomials crack Wilson's quotient at any p^n","Wilson quotient via Fermat sums: one polynomial fits all p^n","Fermat sums unlock Wilson's quotient modulo any p^n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001317,"raw_usage":{"total_tokens":5161,"prompt_tokens":661,"completion_tokens":4500,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":4418}},"tokens_in":405,"tokens_out":4500,"duration_ms":35156,"temperature":1.0,"reasoning_tokens":4418,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:30:44.337128+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Lerch,Zur Theorie des Fermatschen Quotienten ap−1−1 p =q(a), Math","cited_arxiv_id":null,"evidence_quote":"Supplies the base congruence W_p ≡ Σ q_p(a) mod p (Lerch), which is the ℓ=1 first step of the ladder."},{"cited_title":"Lagrange,D´ emonstration d’un th´ eor` eme nouveau concernant les nombres premiers, Nouv","cited_arxiv_id":null,"evidence_quote":"Supplies the polynomial relation whose product form generates the key identity of Lemma 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Bell polynomials used to extract coefficients and express symmetric polynomials in terms of power sums."},{"cited_title":"Comtet,Advanced Combinatorics","cited_arxiv_id":null,"evidence_quote":"Provides the Newton-identity and Bell-polynomial machinery, including the form of the σ^★_k polynomials used in the recurrence."}],"review_version":1}