{"id":"c648e578-efca-4313-9b4e-eaa099cf5880","arxiv_id":"2501.01178","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New congruences modulo 9 and 27 for Lehmer-Euler numbers are proved, residue periodicity modulo higher powers of 3 is tabulated and conjectured, and a new polynomial sequence is connected to Euler and central factorial numbers.","lead":"This paper proves congruence rules and periodic residue patterns for Lehmer-Euler numbers, a 1935 generalization of Bernoulli and Euler numbers, and introduces a new polynomial sequence tied to Euler and central factorial numbers. These results give structural information about a classical number sequence and supply a testable conjecture for how its residues repeat modulo powers of 3.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's proof uses an unproved binomial congruence modulo 27; the stated all-powers version is false, but the needed case is true and fixable.","rationale":"The reader's weakest assumption correctly identifies the unproved binomial congruence after Lucas's theorem as the point where Theorem 4's proof is incomplete. I checked the arithmetic and found that the stronger stated claim, for all powers of 3 and sufficiently large n, is not correct: n = 5, k = 0, ℓ = 1 gives a failure modulo 243. However, the paper only needs the mod-27 case, and that case is true via an elementary factor-pairing argument, so the theorem is very likely correct and the gap is fixable by inserting a short lemma. The mismatches found in the displayed computations of S0, S1, S2 for small n are handled by the authors' separate inspections of n = 1, 2, 3 and are consistent with the claimed congruence pattern. Theorems 3, 5, and 6 have cleaner proofs, and the numerical values of W_{3n} support the main congruences. Because the central proof has a real but repairable omission, the reader's CONDITIONAL verdict remains appropriate; I would not move the verdict. The secondary incompleteness in Proposition 2 reinforces CONDITIONAL rather than ACCEPT.","tokens_in":12898,"tokens_out":24609,"duration_ms":218809,"concrete_test":"Verify the lemma binom(3A, 3B) ≡ binom(A, B) (mod 27) for all 0 ≤ B ≤ A ≤ 30 by direct integer computation, and simultaneously check the factor-pairing identity (3i-1)(3i-2) − (3i-9b-1)(3i-9b-2) = 27(2bi − 3b^2 − b). If any pair fails modulo 27, then the induction in Theorem 4 collapses; if the identity holds, it supplies the missing proof and confirms the needed mod-27 case for the recurrences for W_{9n}, W_{9n+3}, W_{9n+6}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is in the proof of Theorem 4 (Section 3). Immediately after quoting Lucas's theorem, the authors assert that binom(9n, 9k+3ℓ) ≡ binom(3n, 3k+ℓ) modulo 3^j for every j ≥ 1 once n is large enough. This is not proved, and as stated it is false: for n = 5, k = 0, ℓ = 1, we have binom(45,3) = 14190 and binom(15,1) = 15, whose difference 14175 is congruent to 81 modulo 243, so the mod-3^5 claim fails even for a moderately large n. The proof of W_{9n}, W_{9n+3}, W_{9n+6} only needs the mod-27 case, and that case is in fact true: using (3m)! = 3^m m! ∏_{i=1}^m (3i-1)(3i-2), one can show that shifting a factor by 3b changes (3i-1)(3i-2) by a multiple of 27, so the ratio of the two binomial coefficients is 1 mod 27. But because this proof is omitted, the induction in Theorem 4 is incomplete exactly where it needs its sharpest input: the vanishing of the mixed sums S1 and S2 modulo 27 depends on the mod-27 congruence holding without extra hypotheses. The residual risk is not that the needed congruence is false, but that the paper does not supply the argument, and the overstrong assertion suggests the authors may not have isolated the true range of validity. Proposition 2 additionally omits most cases of its asserted periodicity pattern, but that is secondary to Theorem 4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Lehmer-Euler numbers W_n, defined in 1935 by Lehmer via cubic roots of unity. The authors prove recurrence, explicit, and determinant formulas (Theorems 1 and 2), and establish congruence properties modulo 9 and 27 (Theorems 3 and 4). They then tabulate periodic residue patterns modulo powers of 3 and propose Conjecture 1 generalizing the period. The paper also introduces incomplete Lehmer-Euler numbers (Section 4), higher-order Lehmer-Euler numbers (Section 5), and a new polynomial sequence Delta(x,k) whose identities with Euler numbers and central factorial numbers are proved in Theorems 5 and 6. The main claimed new arithmetic results are the modulo 27 congruences and the conjectured periodicity.","tokens_in":13246,"tokens_out":3268,"duration_ms":29547,"significance":"If the proof gaps are repaired, Theorems 3 and 4 give a clean congruence pattern for Lehmer-Euler numbers modulo powers of 3, and the paper collects useful recurrence and determinant identities. The new polynomial sequence and its identities with central factorial numbers (Theorems 5 and 6) are proven in full and appear correct. However, the main arithmetic theorem, Theorem 4, currently depends on an unproved and, as stated, false binomial congruence assertion, so the central claim is not yet established in the written form. The paper contains reproducible definitions and several complete induction proofs, which is a strength.","major_comments":[{"comment":"After quoting Lucas's theorem, the proof states: 'In fact, these congruences hold for (mod 3^l) (forall l >= 1) if n is enough large.' This assertion is not proved and is false as stated. For n = 5, k = 0, and ell = 1, the difference binom(45,3) - binom(15,1) = 14190 - 15 = 14175 is not divisible by 3^5 = 243. The subsequent derivation of the congruences for W_{9n}, W_{9n+3}, and W_{9n+6} modulo 27 relies on the vanishing of the mixed sums S1 and S2 modulo 27, which depends on the mod-27 case of this binomial congruence. Since this is the key input to Theorem 4, the proof is incomplete. The mod-27 case may be true and provable, for instance by writing (3m)! = 3^m m! prod_{i=1}^m (3i-1)(3i-2), but that argument is not supplied in the paper.","section":"Section 3, proof of Theorem 4"},{"comment":"The proof of Proposition 2, which asserts periodic residue patterns modulo 3^4 and 3^5, explicitly says 'Other identities are similarly shown and their proofs are omitted.' The displayed congruences for W_{27n+r} modulo 81 and the longer pattern modulo 243 are not derived from Theorem 4 or from any other statement in the paper. If Proposition 2 is intended as a theorem, these cases need either full proofs or a clear statement that they are computational observations. As written, the assertion is not established, and it is load-bearing for Conjecture 1.","section":"Section 3.1, proof of Proposition 2"}],"minor_comments":[{"comment":"The second congruence is written as W_{9n+3} = (-1)^{n-1} without a modulus symbol; since the proof only establishes congruence modulo 27, it should be written as W_{9n+3} congruent to (-1)^{n-1} (mod 3^3).","section":"Theorem 4 statement"},{"comment":"The line 'W_{9n+6} congruent to -1 - 24 - 3 congruent to -1 (mod 3^3)' appears immediately after the computation for W_{9n+6} and seems to be a stray or erroneous remark; it should be removed or replaced with a correct intermediate congruence.","section":"Section 3, proof of Theorem 4, W_{9n+6} part"},{"comment":"The heading 'Lehmer-Euler numbers modulo powers of three' contains the typo 'Lemer-Euler numbers'; it should be 'Lehmer-Euler numbers'.","section":"Section 3.1 heading"}],"recommendation":"major_revision","confidential_remarks":"The central proof gap is real and specifically located: the asserted all-powers binomial congruence is false as stated, and the proof of Theorem 4 does not separately justify the needed mod-27 case. The theorem itself appears likely correct and repairable, so I do not recommend rejection. The paper's scope fits a number theory journal, and the later identities in Sections 6 seem sound; the revision should focus on completing the proof of Theorem 4 and clarifying the status of Proposition 2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper has genuine new results on Lehmer-Euler numbers, but the proof of the mod-27 theorem has a gap that needs fixing before publication. The main claims look true and are worth refereeing.\n\nWhat's actually new: the mod 9 and mod 27 congruences (Theorems 3 and 4), the periodicity table in Proposition 2, and the polynomial Delta(x,k) identities (Theorems 5 and 6). The methods are standard — generating functions, Lucas's theorem, induction — but the results are legitimate extensions, not rehashes of earlier work. Theorem 3 is clean and complete. The induction proofs for Theorems 5 and 6 are solid, and they give a neat connection between Euler numbers and central factorial numbers.\n\nThe soft spots are in Section 3. In the proof of Theorem 4, the authors assert that binom(9n, 9k+3ℓ) ≡ binom(3n, 3k+ℓ) modulo 3^j for all j≥1 once n is large enough. That's false as stated: take n=5, k=0, ℓ=1; the difference is 14175, which is 81 mod 243, so the mod-3^5 version fails. What the proof actually needs is the mod-27 case, and that case is true, but the paper doesn't prove it. The stress-test note sketches a valid argument using the product formula for factorials; that argument should be inserted. Without it, the induction in Theorem 4 relies on an assertion that is both unproved and overstrong. Also, Proposition 2's proof explicitly omits most cases, saying 'Other identities are similarly shown.' For a table of periodicity at four different moduli, that's a lot of unverified content. I'd want the cases filled in or a code snippet verifying them.\n\nMinor quibbles: the 'In fact' refinements in the Theorem 4 proof (e.g., the sharper congruences for n≥n0) are asserted without proof and are not needed for the main claims. The paper could drop them safely. On the positive side, the citations look appropriate — the authors cite the relevant literature, including their own prior work, without over-reliance.\n\nWho this is for: number theorists working on Bernoulli/Euler-type numbers and their congruences. The impact is narrow but real. The gaps are fixable, and the stress-test's counterexample doesn't sink the main theorem. I'd send this to a competent referee and ask for a repaired proof of Theorem 4 and a fuller Proposition 2.","headline":"Real new congruences for Lehmer-Euler numbers, but Theorem 4's proof needs a filled-in binomial congruence argument; the result survives.","tokens_in":13753,"tokens_out":2626,"would_cite":false,"duration_ms":21981,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B68","11A07","05A19"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lehmer-Euler numbers have periodic residues modulo powers of three","keywords":["Lehmer-Euler numbers","Euler numbers","Bernoulli numbers","congruences","Lucas's theorem","binomial coefficients","periodicity","central factorial numbers"],"falsifier":"Direct computation of $W_{3n}$ for $n = 0, 1, \\dots, 50$ using the recurrence $W_{3n} = -\\sum_{k=0}^{n-1} \\binom{3n}{3k} W_{3k}$ and reduction modulo 27 can settle the main theorem's claim: it must yield the sequence $1, -1, -8, -1, 1, 8$ (mod 27) with period 6. Additionally, testing the asserted binomial congruence for specific pairs, say $n = 10$ and each $k$, would show whether the gap in the proof is real.","tokens_in":12649,"feed_emoji":"🔢","tokens_out":10880,"duration_ms":83158,"temperature":0.7,"pith_summary":"This paper studies Lehmer-Euler numbers, the class of numbers defined in 1935 through cubic roots of unity as a generalization of Euler and Bernoulli numbers, and proves that their nonzero entries obey strict periodic congruences modulo powers of three. The main proved result is that modulo 27, the nonzero sequence $W_{3n}$ cycles through the residues $1, -1, -8, -1, 1, 8$ with period 6. It also proves $W_{3n} \\equiv (-1)^n \\pmod{9}$, gives recurrence and explicit formulas for the numbers, and introduces incomplete and higher-order analogues. The paper conjectures the full rule: if $3n \\equiv 3m \\pmod{2 \\cdot 3^k}$, then $W_{3n} \\equiv W_{3m} \\pmod{3^{k+1}}$.","feed_headline":"Modulo 27, Lehmer-Euler numbers repeat every six entries","feed_subtitle":"The paper proves the period-6 pattern modulo 27 and conjectures the rule for all powers of three.","key_machinery":"The load-bearing object is the Lehmer-Euler sequence itself, with generating function $3/(e^t + e^{\\omega t} + e^{\\omega^2 t}) = (\\sum_{l \\geq 0} t^{3l}/(3l)!)^{-1}$, so that $W_n$ is nonzero exactly when $3$ divides $n$. The congruence proofs run through the recurrence $W_{3n} = -\\sum_{k=0}^{n-1} \\binom{3n}{3k} W_{3k}$, which expresses each term as an integer linear combination of earlier terms. The proof evaluates the appearing binomial-coefficient sums via the identity $\\sum \\binom{3n}{3k+\\ell} x^{3k+\\ell} = \\frac{1}{3}\\sum_{j=0}^2 \\omega^{\\ell j}(1+\\omega^j x)^{3n}$, turning them into explicit powers of $3$ times expressions in $\\sqrt{-3}$. Lucas's theorem gives these sums modulo $3$, and the paper asserts a stronger unproved binomial congruence modulo arbitrary powers of $3$ to move the congruences up to modulo 27.","core_discovery":"The central claim is Theorem 4: for every $n \\geq 0$, the congruences $W_{9n} \\equiv (-1)^n$, $W_{9n+3} \\equiv (-1)^{n-1}$, and $W_{9n+6} \\equiv 8(-1)^{n-1}$ hold modulo 27. From these, the paper derives the period-6 pattern of $W_{3n}$ modulo 27 and, by an analogous argument that it sketches, the corresponding residue patterns modulo $3^4$ and $3^5$. The paper also proposes Conjecture 1 as the general law, asserting that the residue of $W_{3n}$ modulo $3^{k+1}$ depends only on the class of $n$ modulo $2 \\cdot 3^k$. In addition, the paper establishes a new identity expressing Euler numbers in terms of central factorial numbers through a polynomial sequence defined by a two-term recurrence, and it extends the recurrence, determinant, and explicit-formula toolkit to incomplete and higher-order Lehmer-Euler numbers.","pith_inferences":["The unproved binomial congruence $\\binom{9n}{9k+3\\ell} \\equiv \\binom{3n}{3k+\\ell} \\pmod{3^j}$ for all $j$ when $n$ is large is the delicate step; if it can be established by p-adic methods, the proof of Theorem 4 would be fully rigorous and the same method might prove Conjecture 1.","The observed period doubling suggests that $W_{3n}$ is 3-adically well-behaved: the residue modulo $3^{k+1}$ depends on $n$ modulo $2 \\cdot 3^k$, which is consistent with a 3-adic analytic continuation whose Taylor coefficients encode the roots-of-unity sums.","Applying the same generating-function identity with $r$-th roots of unity to the higher-order numbers $W_{r,n}^{(\\alpha)}$ defined in Section 5 may yield analogous periodic congruences modulo powers of $r$."],"forward_implications":["If Theorem 4 is correct, the residue of $W_{3n}$ modulo 27 is determined entirely by $n$ modulo 6, reproducing the explicit six-entry pattern in the paper.","If Conjecture 1 is true, the congruence class of $W_{3n}$ modulo any $3^{k+1}$ is a function only of $n$ modulo $2 \\cdot 3^k$, mirroring the classical result for Euler numbers modulo powers of two but with period doubled at each level.","The identity with central factorial numbers provides a fresh explicit formula for Euler numbers in terms of the polynomial sequence $\\Delta(x,k)$, which may be useful for studying divisibility of Euler numbers.","The incomplete and higher-order Lehmer-Euler numbers inherit the same recurrence and determinant structure, so the congruence machinery can be applied to those families as well."],"supporting_citations":[{"why":"Defines the Lehmer-Euler numbers and gives their generating function, which is the starting point of the whole paper.","marker":"[13]"},{"why":"Supplies the recurrence formula used in the proof of the congruences and the determinant expressions.","marker":"[1]"},{"why":"Provides further properties of the Lehmer-Euler numbers, including recurrence and representation formulas on which the present work builds.","marker":"[10]"},{"why":"Lucas's theorem is used to reduce binomial coefficients modulo 3, the foundation of the congruence arguments.","marker":"[16]"},{"why":"The classical Euler-number congruence modulo powers of two that motivates the conjectured periodicity for the Lehmer-Euler numbers.","marker":"[24]"}],"fun_headline_variants":["Lehmer-Euler numbers period-6 mod 27","Mod 27, Lehmer-Euler numbers repeat every six","Six-step cycles in Lehmer-Euler numbers mod 27","Lehmer-Euler numbers cycle modulo 27 with period 6","Period six proven for Lehmer-Euler mod 27"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the modulo-27 congruences relies on the unproved assertion that $\\binom{9n}{9k+3\\ell} \\equiv \\binom{3n}{3k+\\ell}$ modulo every power of three once $n$ is large enough, which is stronger than Lucas's theorem and is stated without proof.","fun_headline_variants_meta":{"raw":{"variants":["Lehmer-Euler numbers period-6 mod 27","Mod 27, Lehmer-Euler numbers repeat every six","Six-step cycles in Lehmer-Euler numbers mod 27","Lehmer-Euler numbers cycle modulo 27 with period 6","Period six proven for Lehmer-Euler mod 27"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1504,"prompt_tokens":815,"completion_tokens":689,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":602}},"tokens_in":431,"tokens_out":689,"duration_ms":6099,"temperature":1.0,"reasoning_tokens":602,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:33:26.016901+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Direct computation of $W_{3n}$ for $n = 0, 1, \\dots, 50$ using the recurrence $W_{3n} = -\\sum_{k=0}^{n-1} \\binom{3n}{3k} W_{3k}$ and reduction modulo 27 can settle the main theorem's claim: it must yield the sequence $1, -1, -8, -1, 1, 8$ (mod 27) with period 6. Additionally, testing the asserted binomial congruence for specific pairs, say $n = 10$ and each $k$, would show whether the gap in the proof is real.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Lehmer-Euler numbers and gives their generating function, which is the starting point of the whole paper."},{"cited_title":"Barman and T","cited_arxiv_id":null,"evidence_quote":"Supplies the recurrence formula used in the proof of the congruences and the determinant expressions."},{"cited_title":"Komatsu and R","cited_arxiv_id":null,"evidence_quote":"Provides further properties of the Lehmer-Euler numbers, including recurrence and representation formulas on which the present work builds."},{"cited_title":"Lucas, Th´ eorie des Fonctions Num´ eriques Simplement P´ eriodiques, Amer","cited_arxiv_id":null,"evidence_quote":"Lucas's theorem is used to reduce binomial coefficients modulo 3, the foundation of the congruence arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical Euler-number congruence modulo powers of two that motivates the conjectured periodicity for the Lehmer-Euler numbers."}],"review_version":1}