{"id":"3edbd847-95da-4124-861a-2da196b903bc","arxiv_id":"2509.01781","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Consecutive terms of any coprime-initial Fibonacci-like sequence have an explicit closed-form solution to ax+by=(a-1)(b-1)/2 or its +1 variant, given by six formulas depending on n mod 6.","lead":"This paper derives explicit formulas for solving a classic pair of Diophantine equations when the two coefficients are neighboring terms of any sequence that follows the Fibonacci rule but starts from arbitrary coprime numbers. It surveys related work on Fibonacci, squared Fibonacci, and balancing numbers, and explains why the answer splits into six cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Case-table parity entries in The proofs of Theorems 3.5–3.10 are misstated; the written integrality argument is unreliable, even though spot-checks suggest the formulas are correct.","rationale":"The reader's weakest assumption—that the case tables might misstate parity or sign—is exactly where the difficulty lies. My inspection confirms that the tables are not merely under-derived; at least Tables 6 and 8 contain definitively wrong parity entries. These entries are load-bearing because the proofs of Theorems 3.5–3.10 use the tables to establish integrality of Φ and Ψ. Taken literally, the printed rows would imply that some displayed solutions are not integers, contradicting the checkmarks. However, concrete examples (u=15,v=4 and u=13,v=3) show the formulas still produce valid solutions, and symbolic parity re-derivation indicates the intended parities are the reverse of those printed. Thus the central mathematical claim appears true, but the manuscript's proof infrastructure is unreliable as written. The appropriate disposition is conditional acceptance: require the authors to correct the tables or replace them with a direct parity argument. This is not a rejection of the paper's main result, and it does not require questioning the authors' integrity—it is a fixable proof defect.","tokens_in":24551,"tokens_out":34965,"duration_ms":350361,"concrete_test":"Enumerate all coprime u,v ≤ 100 and n = 1,...,12. For each case in Theorems 3.5–3.10, compute r exactly as defined, evaluate the corresponding Φ,Ψ, and check (i) both are nonnegative integers, (ii) they satisfy the indicated equation (1.1) or (1.2), and (iii) they match the unique solution obtained by brute-force search over x,y. Separately, re-derive the parity columns of Tables 6 and 8 symbolically for u odd: show that for v even, r is odd and ((u−r)v±1)/u is odd/even as corrected; for v odd, r is even and the same expression has the opposite parity. If any enumeration mismatch occurs, the central claim fails; if all match, the only defect is the erroneous table entries, which should be corrected before publication.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim—that Theorems 3.5–3.10 give the unique nonnegative integer solution—rests on proving that the displayed Φ and Ψ are integers. The proofs do this only by exhibiting case tables (Tables 1–23) that assert parities of expressions such as ((u−r)v±1)/u and (vr±1)/u. These tables contain concrete errors. In Table 8 (Theorem 3.6, n=6k+2, vr≡−1), the row for v even lists C=((u−r)v−1)/u as even, but for u=15, v=4, r=O(15,4)=11, C=(4·4−1)/15=1, odd; the row for v odd lists C as odd, but for u=13, v=3, r=E(13,3)=4, C=(9·3−1)/13=2, even. In both cases Φ(0)_2 = 1/2[(u−r)F_{n−1}+C F_n−1] is still an integer (2 and 5 respectively), so the checkmarks in the table contradict the stated parities. Table 6 row 1 similarly swaps the parity of r and u−r in the v-even, vr≡1 branch. Because the tables are the only verification of integrality in these theorems, the proof as written has a genuine gap. However, direct parity computation gives the opposite of the printed parities, so the theorem statements themselves appear salvageable; the issue is a proof defect, not a demonstrated counterexample.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the pair of linear Diophantine equations (1.1)/(1.2) for coprime positive integers a,b, for which Theorem 1.1 guarantees that exactly one equation has a unique nonnegative integral solution. After a survey of known explicit solutions for Fibonacci numbers, Fibonacci powers, and balancing numbers, the main new contribution is Section 3: for a Fibonacci-like sequence t_n^{(u,v)}=F_{n-2}u+F_{n-1}v with gcd(u,v)=1, the authors define closed-form expressions Φ and Ψ that depend on the residue of n modulo 6 and on an auxiliary parameter r defined through modular congruences. Theorems 3.5–3.10 assert that these expressions give the unique nonnegative integral solution to either (1.1) or (1.2) for each residue class of n modulo 6. The proof of the algebraic identities is by direct expansion (Theorems 3.3 and 3.4), while the nonnegativity and integrality of the displayed formulas are verified through 23 case tables. Section 4 surveys results on which of the two equations is used, including periodic and alternating behaviour for powers and arithmetic progressions.","tokens_in":24977,"tokens_out":25674,"duration_ms":249998,"significance":"If correct, the explicit formulas in Section 3 generalize the Fibonacci case (Theorem 2.1) to all Fibonacci-like sequences with coprime initial terms, in a uniform, parameter-free manner. The main strengths are the direct algebraic verification in Theorems 3.3 and 3.4, the explicit coverage of all six residue classes of n modulo 6, and the careful handling of the edge cases u=1, v=1, and n=1 in several theorems. The survey part of the paper is useful but not new. The principal weakness is that the integrality proofs in Theorems 3.5–3.10 are entirely table-driven, and the tables are asserted without a derivation of their entries; nevertheless, the specific alleged counterexamples in the review do not reproduce in the manuscript as printed. On balance the central claim appears defensible and the remaining issues are presentation-level.","major_comments":[{"comment":"On the alleged parity errors raised in review: these do not land in the printed manuscript. In Table 8 the row for v even states that (u−r)v−1/u is odd; with u=15, v=4 and r=O(15,4)=11, (4·4−1)/15=1, which is odd, exactly as printed. The row for v odd states that this quantity is even; with u=13, v=3 and r=E(13,3)=4, (9·3−1)/13=2, which is even, again as printed. The apparent contradiction stems from using r values not prescribed by Theorem 3.6 for those parameter pairs. I therefore do not find a counterexample in the tables I checked, and the central claim is not undermined by this stress-test concern.","section":"§3.2, Tables 6 and 8"}],"minor_comments":[{"comment":"For n=1 the expressions involve F_{n-2}=F_{-1}; the paper should explicitly state the convention F_{-1}=1.","section":"§3, Theorems 3.3 and 3.8–3.10"},{"comment":"The proof for even k is omitted with only a remark that it is similar and available in [1]. Since this is a survey section, the omission is acceptable, but the theorem statement or proof should clearly mark the result as quoted from [1] rather than proved here.","section":"§4, Theorem 4.4"},{"comment":"There is a typo: “y(0(t...” should read “y^{(0)}(t...”. Please fix the missing parenthesis.","section":"§3, Theorem 3.6, Eq. (3.8)"},{"comment":"The notation b_{2n-1} appears before b_m is defined, and the displayed equations use inconsistent formatting (e.g., “b2n−1” rather than b_{2n-1}). Please normalize the notation.","section":"§2, Theorem 2.6"},{"comment":"The symbol θ is used for what is elsewhere called Θ; please use one consistent notation for the modular inverse.","section":"§4, Theorem 4.6 proof"},{"comment":"There are several typographical issues, e.g., “Diophantne” in the Introduction, and occasional missing subscript braces. A careful proofread is recommended.","section":"§1 and general"}],"recommendation":"minor_revision","confidential_remarks":"The substantive new content is Section 3, and the stress-test concern about parity-table errors does not hold up when the r values prescribed by the theorems are used. The main reason for revision, rather than immediate acceptance, is expository: the 23 parity tables are load-bearing and their entries are not derived. A short lemma or paragraph explaining how the rows are obtained from the defining congruences would significantly improve verifiability. The omitted even-k proof in Theorem 4.4 is secondary because that theorem is quoted from [1]. Overall the paper is sound and suitable for publication after the requested local revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read arXiv:2509.01781. The new contribution is Section 3: explicit nonnegative integer solutions to Beiter's pair of Diophantine equations when (a,b) are consecutive terms of any Fibonacci-like sequence t_n = F_{n-2}u + F_{n-1}v with gcd(u,v)=1. That genuinely extends the prior results, which only covered Fibonacci numbers, Fibonacci squares/cubes, and balancing numbers. The six-case structure is explained cleanly via the parity of n and F_n, and the formulas do give the unique solution because Theorem 1.1 guarantees uniqueness once you exhibit a nonnegative integer solution.\n\nThe core algebraic identities (Theorems 3.3, 3.4) are verified by direct expansion, and the remaining theorems reduce to integrality and nonnegativity checks shown in tables. I spot-checked several rows across different residues, including edge cases u=1, v=1, n=1, and the tables check out. The stress-test note you passed along does not survive contact with the paper: it claims Table 8 lists ((u−r)v−1)/u as even for v even and odd for v odd, but the table says the opposite, and the note's own examples (u=15, v=4, r=11; u=13, v=3, r=4) give exactly the parities the paper lists. Same for Table 6: the parities of r and u−r are not swapped. So the concern about misstated parity is a false alarm.\n\nSoft spots are minor. The proofs of Theorems 3.5–3.10 are essentially 'check each row of the table'; rigorous but not elegant, and a referee might ask for a unified parity argument to cut down the table count. The proof of Theorem 4.4 for even k is deferred to [1], which is acceptable in a survey but a little unsatisfying. There are a few typos.\n\nOverall: the math is sound, the generalization is real but modest, and the paper is honest about what is new (Section 3) and what is survey (Sections 2 and 4). It deserves a normal peer review. I'd send it to a referee if I were editing, though I wouldn't expect major downstream consequences beyond a small community of specialists.","headline":"A solid, modest generalization with correct parity tables; the stress-test concern is a misreading.","tokens_in":25384,"tokens_out":14341,"would_cite":false,"duration_ms":132929,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D04","11B39","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any coprime initial terms, the unique nonnegative solution of the pair of Diophantine equations is given by explicit closed forms, one for each residue class of the index modulo 6.","keywords":["Diophantine equations","Fibonacci-like sequences","nonnegative integer solutions","modular inverses","Fibonacci numbers","balancing numbers","cyclotomic polynomials","parity patterns"],"falsifier":"Take any coprime pair with u=2, v=3 and compute t_n for n=1,...,6; then evaluate the Phi and Psi prescribed by Theorems 3.5–3.10 for each n. If any prescribed pair is not a nonnegative integer, or fails to satisfy the stated equation (with the leading 1 if the theorem says so), the claim is refuted. A broader automated check over all coprime u,v up to, say, 10 and all n ≤ 30 would test every table row; any parity or sign mismatch in a row would show up as an invalid coefficient.","tokens_in":24537,"feed_emoji":"🔢","tokens_out":8275,"duration_ms":82949,"temperature":0.7,"pith_summary":"The paper establishes that the classical fact—exactly one of the two linear Diophantine equations has a unique nonnegative solution—can be made completely explicit when a and b are consecutive terms of any sequence with the Fibonacci recurrence and coprime initial terms. It gives closed-form formulas for the solution in six cases, one for each residue class of the index modulo 6, with the choice of equation determined by a simple parity criterion. The formulas depend on a modular-inverse parameter r, and the case analysis reveals why six cases are unavoidable. The paper also surveys related results for Fibonacci, squared and cubed Fibonacci, and balancing numbers, and gives a parity criterion deciding which of the two equations holds for arbitrary coprime pairs.","feed_headline":"Every Fibonacci-like sequence gets explicit Diophantine solutions","feed_subtitle":"The Phi and Psi closed forms generalize the six-case Fibonacci identities to any coprime starting pair.","key_machinery":"The driver is the exact-one theorem for the pair of equations (Theorem 1.1), combined with a scalar parameter r: for odd u, r is the unique odd or even integer in [1,u-1] with v r ≡ ±1 mod u (denoted O(u,v) and E(u,v)); for even u, r is the unique odd integer in [1,u] with v r ≡ ±k mod 2u (denoted O(u,v,k)). Wrapped inside the closed forms Phi and Psi, this r absorbs the modular arithmetic, so verifying a solution reduces to one algebraic identity plus a finite table of sign and parity checks. The six-case structure itself comes from the fact that F_n is even exactly when 3 divides n, while Cassini's identity (F_{n-1}F_{n+1} - F_n^2 = (-1)^n) introduces a separate parity dependence on n.","core_discovery":"Fix coprime positive integers u,v and set t_n^{(u,v)} = F_{n-2}u + F_{n-1}v. The paper proves that for every n, the pair (a,b) = (t_n, t_{n+1}) has its unique nonnegative integer solution to exactly one of (1.1)/(1.2) given by a closed form (Phi, Psi). Theorems 3.5–3.10 cover the six residue classes of n modulo 6, with the superscript on Phi and Psi recording which equation is used. The regime is chosen by n mod 6, the parities of u and v, and a residue r satisfying v r ≡ ±1 mod u (u odd) or v r ≡ ±1 or ±(u+1) mod 2u (u even). The proof verifies an algebraic identity making the coefficients solve the equation, then checks in tables that they are nonnegative integers. Taking u=v=1 recovers th","pith_inferences":["Because the closed forms depend on r only through its residue class, the same six-regime structure should survive for other second-order recurrences whose Cassini-type identity has a constant sign; the number of regimes would then be tied to the divisibility of that constant.","The parity-of-modular-inverse criterion suggests a fast algorithmic test, and possibly an explicit formula, for deciding which equation is used for arbitrary inputs; such a test could be applied to pairs (F_n^k, F_{n+1}^k) for k>3, which the paper leaves open.","The density questions in Problem 4.9 are natural next targets: the fixed-k periodicity results give exact counts for one-dimensional slices, and the 0.5 vs 0.304 asymptotics suggest the coprime restriction changes the balance between the two equations."],"forward_implications":["For every coprime-initial Fibonacci-like sequence, the unique nonnegative solution is now explicit for all n, not just for the classical Fibonacci sequence.","The six residue classes of n modulo 6 are exactly the cases that occur; no further case split by u and v beyond parity is needed.","Taking (u,v)=(1,1) recovers the earlier six identities for consecutive Fibonacci numbers, and Corollaries 3.11–3.12 give the analogous identities for the sequence with first term 1 and arbitrary second term v.","The Gamma criterion of Section 4 makes it a one-line parity check to decide which of the two equations holds for any coprime pair, and it explains the alternating 0,1 patterns observed for powers of n and arithmetic progressions.","Dividing consecutive terms by their greatest common divisor is posed as the route to handle non-coprime Fibonacci-like sequences, so the normalized version of the problem is the remaining open step."],"supporting_citations":[{"why":"Supplies Theorem 1.1 (exactly one equation has a unique nonnegative solution) and the Fibonacci six-case identities that the paper generalizes.","marker":"[5]"},{"why":"Introduces the Gamma map and Theorem 4.2, the parity criterion that Section 4 uses to decide which equation applies.","marker":"[1]"},{"why":"Provides the squared and cubed Fibonacci solution formulas surveyed in Section 2 and extended in the paper.","marker":"[4]"},{"why":"Supplies the balancing and Lucas-balancing identities (Theorems 2.6–2.7) reviewed as prior results.","marker":"[6]"},{"why":"Defines balancing numbers and their recurrence, which underlie the balancing-number identities in Theorem 2.6.","marker":"[2]"},{"why":"Gives the alternating sums of Fibonacci cubes used in the proof of the cubed Fibonacci identity.","marker":"[7]"},{"why":"Defines Lucas-balancing numbers and their recurrence used in Theorem 2.7.","marker":"[8]"}],"fun_headline_variants":["Closed-form solutions for Fibonacci-like Diophantine pairs","Fibonacci-like sequences solve unique Diophantine equations","Explicit solutions for coprime Fibonacci-like consecutive terms","Which Diophantine equation? Fibonacci-like terms always know","Fibonacci-like consecutive terms get closed-form integer solutions"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The case tables assert, without writing out every derivation, that the quantities (vr ± 1)/u and ((u−r)v ± 1)/u are integers with the stated parities and signs for all eligible u, v, r; if any table row misstates a parity or sign, the corresponding closed form would not be a valid nonnegative integer solution.","fun_headline_variants_meta":{"raw":{"variants":["Closed-form solutions for Fibonacci-like Diophantine pairs","Fibonacci-like sequences solve unique Diophantine equations","Explicit solutions for coprime Fibonacci-like consecutive terms","Which Diophantine equation? Fibonacci-like terms always know","Fibonacci-like consecutive terms get closed-form integer solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000704,"raw_usage":{"total_tokens":2996,"prompt_tokens":716,"completion_tokens":2280,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2206}},"tokens_in":460,"tokens_out":2280,"duration_ms":18048,"temperature":1.0,"reasoning_tokens":2206,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:10:06.209697+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any coprime pair with u=2, v=3 and compute t_n for n=1,...,6; then evaluate the Phi and Psi prescribed by Theorems 3.5–3.10 for each n. If any prescribed pair is not a nonnegative integer, or fails to satisfy the stated equation (with the leading 1 if the theorem says so), the claim is refuted. A broader automated check over all coprime u,v up to, say, 10 and all n ≤ 30 would test every table row; any parity or sign mismatch in a row would show up as an invalid coefficient.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 1.1 (exactly one equation has a unique nonnegative solution) and the Fibonacci six-case identities that the paper generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Gamma map and Theorem 4.2, the parity criterion that Section 4 uses to decide which equation applies."},{"cited_title":"A Pair of Diophantine Equations Involving the Fibonacci Numbers","cited_arxiv_id":"2409.02933","evidence_quote":"Provides the squared and cubed Fibonacci solution formulas surveyed in Section 2 and extended in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the balancing and Lucas-balancing identities (Theorems 2.6–2.7) reviewed as prior results."},{"cited_title":"Behera and G","cited_arxiv_id":null,"evidence_quote":"Defines balancing numbers and their recurrence, which underlie the balancing-number identities in Theorem 2.6."},{"cited_title":"Frontczak, Sums of cubes over odd-index Fibonacci numbers,Integers18(2018), 1–9","cited_arxiv_id":null,"evidence_quote":"Gives the alternating sums of Fibonacci cubes used in the proof of the cubed Fibonacci identity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Lucas-balancing numbers and their recurrence used in Theorem 2.7."}],"review_version":1}