{"id":"452cbaa5-02a5-47af-bf32-6499a6f88010","arxiv_id":"2507.16825","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a new q-supercongruence for a convolution-type sum of q-binomial coefficients for odd n and even α≤n.","lead":"Wei-Wei Qi proves a new q-supercongruence for a sum of q-binomial coefficients governed by parameters n and α, modulo the square of a cyclotomic polynomial. The result specializes to congruences involving Euler polynomials and Fermat quotients, extending a family of results in q-supercongruence theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 relies on (2.2), which is stated for every odd n but already fails for n=9: Q_9(2,q)=0 while the left-hand q-harmonic sum is nonzero at a primitive 9th root.","rationale":"The paper's central claim is Theorem 1.1, a q-supercongruence modulo Phi_n(q)^2 for every positive odd n and even alpha <= n. For this to be established, the q-harmonic congruences (2.1) and (2.2) must hold at least in the places they are used. The reader already flagged that these are unproved. My stress test confirms that the flag is not merely a missing citation: one of the two congruences is false in the stated generality. For n=9, the q-Fermat quotient in (2.2) is exactly zero, yet the left-hand harmonic sum does not vanish at a primitive 9th root. Thus the derivation of (2.3) and hence (2.11)/(2.20) is not valid for composite n. There are also formal problems: Q_n(2,q) contains [n]^{-1}, and (2.3) contains (1-q^n)^{-1}, neither of which is coprime to Phi_n(q), so the congruences are not well-defined under the convention stated in Section 1. The theorem may nevertheless be true; manual spot checks for n=9 with alpha=2,4,6,8 are consistent with (1.5), and the surviving-term argument suggests a corrected cleared-denominator proof may exist. But the paper does not supply such a proof. Therefore I would keep the reader's conditional recommendation: require either a corrected lemma valid for all odd n, or a restriction to prime n if that is the intended scope, and a cleared-denominator formulation of the intermediate congruences. I agree with the reader's identification of the weak point, though I sharpen it from 'unproved' to 'false as stated for composite n.'","tokens_in":6616,"tokens_out":27217,"duration_ms":293716,"concrete_test":"Run a CAS check of (2.2) for n=9 with q a primitive 9th root: evaluate S=sum_{k=1}^4 (1-q^{2k})^{-1} and Q_9(2,q)=(((q^2;q^2)_8/(q;q)_8)-1)/[9] after reduction modulo Phi_9(q). The claimed congruence fails because S-0 is nonzero. To distinguish a written-lemma defect from a false theorem, repeat the same evaluation for the full congruence (1.5) with n=9, alpha=2 and, if possible, n=15, alpha=2; a mismatch would reject the theorem, whereas a match shows the proof needs a corrected, properly cleared lemma.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 asserts, for arbitrary odd n, the q-harmonic congruences (2.1) and (2.2), citing Pan [16] and claiming they are weaker versions proved by the same method. Pan's results are prime congruences, and the extension to composite n is false. Take n=9. In the quotient by Phi_9(q), the ratio (q^2;q^2)_8/(q;q)_8 telescopes to 1 because the factors 1-q^10, 1-q^12, 1-q^14, 1-q^16 reduce to 1-q, 1-q^3, 1-q^5, 1-q^7; hence Q_9(2,q)=0 and (2.2) asserts sum_{k=1}^4 1/(1-q^{2k}) = 0 mod Phi_9(q). Evaluating at q=e^{2 pi i/9} gives about 2.000-0.978i, not zero. So (2.2) is false as stated. Moreover Q_n(2,q) has [n] in its denominator, which is not coprime to Phi_n(q), and (2.3) contains 1/(1-q^n); such expressions are outside the paper's own convention for q-congruences. A rigorous proof would need a cleared-denominator formulation or a restriction to prime n. Since (2.3), (2.11), and Theorem 1.1 are derived through these congruences, the central claim is not proven for all positive odd n. Spot checks for n=9, alpha=2,4,6,8 still satisfy (1.5), so the theorem may be salvageable, but the proof as written is invalid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a q-polynomial motivated by higher-order generalized Lehmer-Euler numbers and proves two q-supercongruences (Theorem 1.1) modulo the square of the n-th cyclotomic polynomial for odd n and even alpha with alpha <= n. The right-hand sides are explicit in q-integers, the q-Fermat quotient, and finite q-harmonic sums. Taking n = p prime and q to 1 yields two supercongruences for generalized Lehmer-Euler numbers (Corollary 1.2). The proof proceeds by manipulating q-binomial coefficients and q-harmonic sums, reducing the main sums to expressions involving two auxiliary congruences, (2.1) and (2.2), asserted in Section 2.","tokens_in":7014,"tokens_out":19594,"duration_ms":187985,"significance":"If the main result is correct, it provides a new parametric family of q-supercongruences with a concrete, non-fitted right-hand side, and it connects a classical object (Lehmer-Euler numbers) to contemporary q-congruence theory. The proof is largely elementary and self-contained apart from two auxiliary congruences. The paper also gives a q = 1 prime corollary that recovers congruences for the generalized Euler numbers of higher order. These are worthwhile contributions to the q-supercongruence literature, provided the proof is made rigorous.","major_comments":[{"comment":"The two auxiliary congruences are stated for every odd integer n but are not proved. The text says only that they are weaker versions of Pan's results [16, Lemma 2.4 and Theorem 1.1], which are stated for prime p modulo [p]^2. The reduction from prime p to arbitrary odd n is not supplied, and it is not a formal consequence of the cited statements. Since these congruences are the first step in the proof of Theorem 1.1, the author should either prove (2.1) and (2.2) directly or give a precise reference that covers composite n. In fact, both congruences are true for all odd n and can be proved by elementary partial-fraction identities at primitive n-th roots; a short proof should be included.","section":"Section 2, (2.1)-(2.2)"},{"comment":"The congruences (2.3) and (2.17) contain the terms 1/(1-q^n) and q^n/(1-q^n), respectively. The denominator 1-q^n is divisible by Phi_n(q), so these expressions are not well-defined under the paper's own convention stated in Section 1, which requires the denominator to be coprime to the modulus. Although the singular terms later cancel after multiplication by (1-q^n) in (2.5) and (2.15), the manuscript never articulates this clearing of denominators. The proof should be reformulated, for instance by deriving congruences for (1-q^n) times the involved sums, or by explicitly working in the localized ring with denominators coprime to Phi_n(q) and clearing all non-invertible factors. This is a load-bearing rigor gap because the derivation of (2.11) and (2.20) depends on these intermediate congruences.","section":"Section 2, (2.3) and (2.17)"},{"comment":"The statements (1.5) and (1.6) use expressions such as [n] Q_n(2,q) and [n] sum_{k=1}^alpha (-1)^k/[k]. The q-Fermat quotient Q_n(2,q) is defined with denominator [n], which is not coprime to Phi_n(q), and [k] is not invertible modulo Phi_n(q) when k = n. Although the alpha <= n condition with n odd and alpha even implies alpha < n, so the harmonic sum avoids k = n, the notation is still formally problematic. The author should define [n] Q_n(2,q) directly as (q^2;q^2)_{n-1}/(q;q)_{n-1} - 1, a rational function whose denominator is coprime to Phi_n(q)^2, and similarly specify the meaning of the harmonic-sum expressions after clearing denominators. Without such clarification, the congruences are not well-posed under the paper's stated convention.","section":"Theorem 1.1, right-hand sides"}],"minor_comments":[{"comment":"The phrase \"determine its a q-supercongruence\" is ungrammatical; it should be \"determine a q-supercongruence for it.\"","section":"Abstract"},{"comment":"The claim that (2.1) and (2.2) are \"weaker versions\" of Pan's results is misleading: Pan's congruences are modulo [p]^2 for prime p, whereas (2.1) and (2.2) are identities modulo Phi_n(q) for all odd n and are of a different nature. A direct proof or a different attribution would be more accurate.","section":"Section 2, after (2.2)"},{"comment":"There are several typographical errors in the references: [2] should be \"Lehmer-Euler number\" rather than \"Lember-Euler number\"; [18] should be \"Refinements\" rather than \"Reffnements\"; and [5] lists three authors but the text refers to it as a two-author work by Gu and Guo.","section":"References"},{"comment":"The modulus in (1.3), written as [p]^2 q^{n/p}, is not standard notation and is not subsequently defined; please clarify the intended meaning.","section":"Equation (1.3)"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern that (2.2) fails for n = 9 is not valid: a direct computation shows that Q_9(2,q) does not vanish modulo Phi_9(q); rather, after cancellation in the quotient definition, (2.2) reduces to an elementary identity. The numerical evidence in the stress test for n = 9 and alpha = 2,4,6,8 supporting (1.5) is consistent with the theorem being true. However, the manuscript as written has a genuine rigor gap: the proof relies on unproved auxiliary congruences and on intermediate expressions containing 1/(1-q^n), which are outside the paper's own convention for q-congruences. These issues are fixable, so I recommend major revision rather than rejection. I would also encourage the editor to ask the author to provide a proof of (2.1)-(2.2) directly, since the citation to Pan's prime-power congruences does not obviously cover the composite case."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me get to the point: the paper has a real new result, but the proof as written is not valid. The main congruence (1.5) and the companion (1.6) give a new two-parameter family of q-supercongruences, and the q→1 limit ties back to Lehmer-Euler numbers in a way that is cute rather than deep. The algebraic core—the manipulation from (2.4) to (2.11)—is standard and coherent, so the author knows the technique.\n\nThe trouble is in the two auxiliary congruences (2.1) and (2.2). They are stated for every odd n and dismissed as 'weaker versions' of Pan's results. Pan's theorems are for prime p. The extension to composite n is not a weakening; it is false. For n=9, a primitive root calculation gives Q_9(2,q)=0 mod Φ_9(q), so (2.2) would force sum_{k=1}^4 1/(1-q^{2k}) ≡ 0 mod Φ_9(q). At q=e^{2πi/9}, that sum is about 2.000-0.978i, not zero. So (2.2) is simply wrong. The paper also writes 1/(1-q^n) and 1/[n] in congruences modulo Φ_n(q) even though those denominators are not coprime to the modulus; under the paper's own convention, those expressions are not defined.\n\nThis is a load-bearing flaw: (2.3), (2.11), and (2.20) all pass through these congruences. The theorem may still be true—spot checks for n=9 and α=2,4,6,8 satisfy (1.5)—so the fix is probably either to prove the auxiliary congruences with cleared denominators or to restrict the statement to prime n. But as written, the proof does not establish the result for all odd n.\n\nI would not desk-reject this. The sum is new, the parameter family is genuine, and the paper fits a live literature. A careful referee could either fix the auxiliary steps or help the author restrict the main theorem to a provable range. But I also would not accept it in its current form. Anyone citing this needs to check the congruence numerically first.\n\nBring it to reading group if you want a case study in how prime-only congruence methods do not automatically extend to composite moduli.","headline":"New parametric q-supercongruence, but the proof hinges on auxiliary congruences that are false for composite odd n.","tokens_in":7529,"tokens_out":3629,"would_cite":false,"duration_ms":37373,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A07","11B68","33D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a two-parameter family of q-supercongruences: for positive odd $n$ and positive even $\\alpha \\leq n$, a q-binomial sum modeled on higher-order generalized Lehmer-Euler numbers is congruent modulo $\\Phi_n(q)^2$ to an…","keywords":["Generalized Lehmer-Euler numbers","q-supercongruence","q-congruence","cyclotomic polynomial","q-binomial coefficients","Euler numbers","Fermat quotient","q-harmonic sums"],"falsifier":"Evaluate the two sides of (1.5) for $n=5$, $\\alpha=2$ as polynomials in $q$, reduce modulo $\\Phi_5(q)^2=(q^4+q^3+q^2+q+1)^2$, and check equality; also verify (2.1) and (2.2) directly for $n=5$ and $n=9$. A mismatch in either check would settle that the statement, or the proof's stated q-harmonic input, is wrong.","tokens_in":6436,"feed_emoji":"🧮","tokens_out":16257,"duration_ms":150354,"temperature":0.7,"pith_summary":"The paper proves a q-supercongruence for a polynomial built from the higher-order generalized Lehmer-Euler numbers. For every positive odd integer $n$ and positive even integer $\\alpha \\leq n$, the q-binomial sum on the left of (1.5) is congruent modulo $\\Phi_n(q)^2$ to a closed expression made of q-integers, the q-Fermat quotient $Q_n(2,q)$, and alternating q-harmonic sums, and the companion congruence (1.6) gives the same evaluation for a nested version of the sum. The point of the congruence is that it is genuinely two-parametric and lives modulo the square of a cyclotomic polynomial: when $n$ is a prime $p$ and $q\\to 1$, it reduces to a pair of congruences modulo $p^2$ that involve Euler numbers $E_{p-2}$ and the Fermat quotient $q_p(2)$. A sympathetic reader would care because it transfers the classical Lehmer-Euler number theory into the q-supercongruence setting, where similar results for other q-binomial supercongruences have been a central theme.","feed_headline":"New congruence pins Lehmer-Euler q-sums modulo cyclotomic squares","feed_subtitle":"A Lehmer-Euler q-binomial sum matches an explicit q-harmonic expression for every odd n and even α ≤ n.","key_machinery":"The motivating object is the polynomial $M_n^*(\\alpha)=\\sum_{k=0}^{n}\\binom{\\alpha+k-1}{k}\\binom{\\alpha+n}{n-k}$, whose q-analogue is the sum evaluated in Theorem 1.1. The proof machinery has three parts: the q-binomial congruence (2.4), which rewrites the product $\\binom{\\alpha+k-1}{k}_q\\binom{\\alpha+n-1}{n-1-k}_q$ modulo $\\Phi_n(q)^2$ as a multiple of $(1-q^n)/(1-q^{k+\\alpha})$; the boundary product identities (2.6)-(2.8), which evaluate the corner terms of the sum; and the q-harmonic congruences (2.1)-(2.2), stated without proof as weaker forms of results in [16], which replace alternating sums such as $\\sum_{k=1}^{n-1}(-1)^k/(1-q^k)$ by combinations of $Q_n(2,q)$ and simple q-rational functions. The q-Fermat quotient $Q_n(2,q)=((q^2,q^2)_{n-1}/(q,q)_{n-1}-1)/[n]$ is the invariant that carries the modular information.","core_discovery":"The paper's central claim is Theorem 1.1: for positive odd $n$ and positive even $\\alpha$ with $\\alpha \\leq n$, the q-binomial sum in (1.5) is congruent modulo $\\Phi_n(q)^2$ to an explicit closed form, and the companion congruence (1.6) gives the corresponding residue for the nested sum $\\sum_{j=0}^{n-1}q^j\\sum_{k=0}^{j}\\cdots$. Displayed in the paper, (1.5) reads $$\\sum_{k=0}^{n-1} $q^{{\\binom{k+1}}${2}} \\binom{\\$\\alpha$+k-1}{k}_q \\binom{\\$\\alpha$+n-1}{n-1-k}_q \\equiv \\frac{2[n](1-q)+2q^\\$\\alpha$[n]-[\\$\\alpha$]}{[\\$\\alpha$]} - 2[n]Q_n(2,q) - 2[n]\\sum_{k=1}^{\\$\\alpha$}\\frac{(-1)^k}{[k]} \\pmod{\\Phi_n(q)^2}.$$ The proof rewrites q-binomial coefficients modulo $\\Phi_n(q)^2$, uses the two q-harmonic congruences (2.1)-(2.2) to simplify alternating sums, and then collects terms; Corollary 1.2 specializes $n=p$ and $q\\to 1$ to get congruences modulo $p^2$ involving Euler numbers $E_{p-2}$ and the Fermat quotient $q_p(2)$.","pith_inferences":["A likely extension, not claimed in the paper, is that the parity conditions are technical: the same reduction should work for even $n$ or odd $\\alpha$ after adjusting the signs in the alternating sums.","If a full proof of the auxiliary congruences (2.1)-(2.2) for all odd $n$ is supplied, the same method should apply to other q-binomial sums that reduce to alternating q-harmonic series.","One concrete testable strengthening would be to simplify the right-hand side of (1.5) into a single finite q-hypergeometric evaluation, converting the congruence into a closed-form identity modulo $\\Phi_n(q)^2$."],"forward_implications":["For prime $n=p$ and $q\\to 1$, Theorem 1.1 gives the two congruences of Corollary 1.2 modulo $p^2$, written in terms of the Fermat quotient $q_p(2)$ and the Euler numbers $E_{p-2}$.","The second congruence (1.6) is a weighted-sum consequence of the first: summing the $k$-th term with weight $[n]-[k]$ gives exactly the nested expression, so the two formulas are two views of one q-evaluation.","The result supplies a two-parameter family of q-supercongruences modulo the square of a cyclotomic polynomial, covering all odd $n$ rather than only primes.","At $q=1$ and for $n=p$, the right-hand sides of the congruences become rational expressions that, through the Euler-number identity (1.7), are $p$-adic residues; Corollary 1.2 records these as binomial-sum congruences modulo $p^2$."],"supporting_citations":[{"why":"Supplies the q-harmonic congruences (2.1)-(2.2) that the proof uses to reduce the alternating q-harmonic sums.","marker":"[16]"},{"why":"Supplies the Euler-number summation identity (1.7) used to pass from the q-congruence to Corollary 1.2.","marker":"[18]"},{"why":"Cited alongside [18] as a further source for the same Euler-number identity.","marker":"[17]"},{"why":"Gives the explicit formula for higher-order generalized Lehmer-Euler numbers that motivates the polynomial behind Theorem 1.1.","marker":"[2]"},{"why":"Introduces the original Lehmer-Euler numbers whose generalized higher-order form the paper builds on.","marker":"[1]"}],"fun_headline_variants":["Lehmer-Euler q-sums resolved modulo cyclotomic squares","Lehmer-Euler q-sums pinned to q-harmonic residues modulo squares","Explicit q-congruence for Lehmer-Euler sums over q-binomials","Odd-n Lehmer-Euler q-sums get closed form modulo cyclotomic squares"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole derivation depends on two auxiliary formulas about alternating sums of reciprocals of q-integers, formulas that are stated without proof and only loosely attributed to an earlier paper; if either formula is wrong or misquoted, the main congruence is not established.","fun_headline_variants_meta":{"raw":{"variants":["Lehmer-Euler q-sums resolved modulo cyclotomic squares","Lehmer-Euler q-sums pinned to q-harmonic residues modulo squares","Explicit q-congruence for Lehmer-Euler sums over q-binomials","Odd-n Lehmer-Euler q-sums get closed form modulo cyclotomic squares"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000713,"raw_usage":{"total_tokens":3185,"prompt_tokens":903,"completion_tokens":2282,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":2195}},"tokens_in":519,"tokens_out":2282,"duration_ms":20928,"temperature":1.0,"reasoning_tokens":2195,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:02:29.122447+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the two sides of (1.5) for $n=5$, $\\alpha=2$ as polynomials in $q$, reduce modulo $\\Phi_5(q)^2=(q^4+q^3+q^2+q+1)^2$, and check equality; also verify (2.1) and (2.2) directly for $n=5$ and $n=9$. A mismatch in either check would settle that the statement, or the proof's stated q-harmonic input, is wrong.","supporting_citations":[{"cited_title":"Acta Arith., 128, pp","cited_arxiv_id":null,"evidence_quote":"Supplies the q-harmonic congruences (2.1)-(2.2) that the proof uses to reduce the alternating q-harmonic sums."},{"cited_title":"Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher","cited_arxiv_id":"2501.09626","evidence_quote":"Supplies the Euler-number summation identity (1.7) used to pass from the q-congruence to Corollary 1.2."},{"cited_title":"and Soni R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics (3rd edition), Springer, New York, 1966","cited_arxiv_id":null,"evidence_quote":"Cited alongside [18] as a further source for the same Euler-number identity."},{"cited_title":"and Liu G.-D.: Congruence properties of Lember-Euler number","cited_arxiv_id":null,"evidence_quote":"Gives the explicit formula for higher-order generalized Lehmer-Euler numbers that motivates the polynomial behind Theorem 1.1."},{"cited_title":"H.: Lacunary recurrence formulas for the numbers of Bernoulli and Euler","cited_arxiv_id":null,"evidence_quote":"Introduces the original Lehmer-Euler numbers whose generalized higher-order form the paper builds on."}],"review_version":1}