{"id":"2d54f27d-5d08-491a-a986-e73c8efc6b31","arxiv_id":"2507.12397","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For the equation x^2 - 2 = y^p, the authors prove the only solutions for p>911 are y = -1, and any nontrivial solution has y > 10^1000.","lead":"The authors prove that for every odd prime p>911, the equation x^2 - 2 = y^p has only the trivial solution y = -1, and that any exception must have y > 10^1000. This reduces a long-open conjecture in number theory to 84 specific cases, so a complete solution may now be within reach.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unconditional claims (Theorems 1.2, 1.4, 1.5) rest on finite computational certificates that are not printed or independently reproducible from the paper; the Table 1 parameter verification is especially delicate because condition (5.37) holds with a margin of about 0.02 for p=919.","rationale":"The reader's weakest assumption and my read identify the same load-bearing point: the proof's finite computational certificates are not independently verified from the printed text. The analytic parts of the paper are detailed and internally consistent: the reduction to Thue equations, the linear-forms setup, the Galois-theoretic observations, and the continued-fraction lower-bound argument all read coherently, and the provision of GitHub code is genuine supporting material. However, the headline result has an unusually tight numerical margin at p=919, where condition (5.37) is satisfied by only about 0.02. This makes the missing verification of Table 1 not a stylistic issue but a logical gap in the written proof. The paper would be complete once the code is pinned and the outputs are either printed or certified by an independent run; until then, CONDITIONAL is the right verdict. My concern does not move the reader's verdict, so I mark it UNCHANGED.","tokens_in":44212,"tokens_out":20634,"duration_ms":205976,"concrete_test":"Recompute every row of Table 1 with rigorous interval arithmetic, using K=⌈K′log y0⌉ and the S1,S2 choices from Remark 5.20, and verify (5.38) and (5.39) at the stated y0 for each prime p in that row's range, together with the strict inequality (5.37) using the stated ρ, μ, L, and K′. If every inequality holds with a positive margin, this step of Theorem 5.1 is sound; if any fails, Theorem 1.4 is unproved for that p.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The unconditional claim that Conjecture 1.1 holds for p>911 depends on three finite computational certificates: (i) the Theorem 5.3 auxiliary-prime intersection proving r=±1 for 17≤p<20000; (ii) the Theorem 5.1 parameter search summarized in Table 1, together with the inequalities (5.38) and (5.39) asserted for each prime in 919≤p≤1951; and (iii) the continued-fraction lower bounds in Section 6. None of these certificates is printed; the paper points to GitHub but gives no commit pin, no full logs, and no explicit verification of the Table 1 inequalities. The most delicate point is (ii): condition (5.37), log(ρ)μLK′

911) and Theorem 1.5 (any nontrivial solution has y > 10^1000). Together with Chen's modular result, these theorems reduce the conjecture to 84 prime exponents 17 <= p <= 911. The paper also contains a detailed Galois-theoretic analysis of the associated Thue polynomials, local observations, and a study of the newforms of level 128.","tokens_in":44551,"tokens_out":10093,"duration_ms":118491,"significance":"If the computational certificates are valid, the paper substantially advances a long-standing folklore conjecture: it settles the conjecture for p <= 13 and for all odd primes p > 911, leaving only 84 prime exponents. The analytic core is presented in unusually complete detail: the Galois group computation, the reduction to one Thue equation once r = ±1, the three applications of Laurent's bounds, and the continued-fraction argument are all readable and checkable. The paper also gives credit to and clarifies the GRH-dependence of earlier work of Bugeaud–Mignotte–Siksek. The GitHub repository is a positive feature. The central weakness is auditability: the unconditional headline theorems rest on three finite computations (the PARI/GP Thue solutions, the auxiliary-prime search behind Theorem 5.3, and the Section 6 continued-fraction bounds) whose complete outputs are not printed, and the Table 1 parameter verification is not shown. These gaps do not appear to reflect circular reasoning or an error in the analytic method, but they currently prevent the paper from being fully verifiable as written.","major_comments":[{"comment":"The proof of Theorem 5.1 terminates by asserting that, for each prime 919 <= p <= 1951, the choices of K', L, R1, R2, mu, and rho in Table 1 satisfy the hypotheses of Propositions 5.26 and 5.27, but the verification of (5.38) and (5.39) is not displayed for any row. This is load-bearing because (5.37) is nearly tight: for the first row, p = 919, log(27.22) * 0.58 * 9 * 26.64 is approximately 459.48, while p/2 = 459.5, a margin of about 0.02. A small rounding error in K', mu, or rho, or a failure of (5.39) at y0 = 10800, would remove the contradiction for p = 919 and Theorem 1.4 would not follow as written. Please print, for every prime in 919 <= p <= 1951, the constants entering Propositions 5.26 and 5.27 and the values of the two inequalities at y0, or provide a commit-pinned script whose single command prints these verifications for all rows of Table 1.","section":"Section 5.8, Table 1, Eqs. (5.36)–(5.39)"},{"comment":"The conclusion that r = ±1 for every 17 <= p < 20000 is essential: it is used to replace the Thue equations (3.2) by the single equation (3.3) and to set b2 = 2 in the linear form (5.16). The proof says that the computation was verified in Sage and that a text file with the auxiliary primes was output, but neither the auxiliary primes for each p nor a checksum or commit identifier is provided. Since the exhaustiveness over all primes in the range is exactly what cannot be checked from the text, please include the auxiliary-prime list for each p (or the script that generates it, together with a certificate or log confirming that the intersection of the sets R_ell(F) is contained in {1,-1} for every p).","section":"Section 5.5, Theorem 5.3"},{"comment":"The unconditional claim for p <= 13 depends on GP/PARI's Thue equation solver being invoked in unconditional mode, but the Thue equations solved, the solver flags, and the resulting solution sets are not shown. The paper itself emphasizes that the default thue function assumes GRH and that the authors could only reach p = 13 unconditionally; therefore the reader needs to see the actual computations in order to confirm that no GRH assumption is hidden. Please include the relevant PARI/GP commands, the list of Thue equations for 3 <= p <= 13, and the output showing that only solutions with r = ±1 and y = -1 occur.","section":"Section 3, proof of Theorem 1.2"},{"comment":"The final lower bounds on y, |a|, and |b| are obtained by a Sage computation of continued fraction expansions of the unique real root theta of f_{1,p}, but the paper prints no per-prime output. The proof of Theorem 1.4 requires showing that the continued fraction quotients satisfy Proposition 6.4 for each prime 919 <= p <= 1951 and that this yields y exceeding the Table 1 values; the proof of Theorem 1.5 requires the analogous verification for 17 <= p <= 911 at the 10^1000 level. As printed, the reader cannot verify either assertion. Please include the per-prime data (for example, the index k and denominator Q_{k+1} obtained for each p) or provide a complete script with pinned dependencies and a log of its output.","section":"Section 6, Theorems 1.4 and 1.5"}],"minor_comments":[{"comment":"In the text, 'mu = 0..508613' should read 'mu = 0.508613'.","section":"Section 5.4, proof of Theorem 5.2"},{"comment":"The statement 'p2 3p−7 2 − 2' is typeset incorrectly; it should be p * 2^{(3p-7)/2} - 2, matching the derivation in (6.4).","section":"Section 6, Proposition 6.4"},{"comment":"The row labels '967 − 997', '1000 − 1200', and '1200 − 1951' mix primes and composite limits, and the endpoint 1200 appears in two rows. Please clarify that each row applies to all primes in the stated interval and avoid the overlap.","section":"Table 1"},{"comment":"The auxilary-prime computation is described for 11 <= p < 20000, while Theorem 5.3 is stated for p >= 17. Please clarify whether p = 11 and p = 13 are included only for completeness or are needed for some downstream argument.","section":"Section 5.5"},{"comment":"The lemma states the assumption y != 1; for nontrivial solutions y >= 23 by Proposition 5.4, so this is harmless, but the hypothesis is presumably meant to be y != -1 for consistency with Definition 1.6.","section":"Section 5.3, Lemma 5.11"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's analytic content is strong and I found no circularity in the main derivation. The only serious obstacle is that the unconditional theorems depend on finite computations whose certificates are not printed and whose Table 1 verification is displayed only by assertion. If the repository already contains per-prime verification logs under a fixed commit, the revision can resolve this quickly; if not, the authors should generate and archive such logs. I would be willing to accept after these certificates are made auditable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about this paper: it is a genuinely new partial result on the Lebesgue–Nagell equation x^2 − 2 = y^p, and its headline theorems rest on computations that are not auditable from the printed page. Read it for the analytic machinery, but treat the numerical certificates as conditional until the code is pinned.\n\nWhat is new: Theorems 1.4 and 1.5 push the known range of the conjecture from p ≥ 1237 down to p > 911, and raise the counterexample lower bound from 10^102 to 10^1000. Theorem 1.2 gives an unconditional proof for p ≤ 13, and the authors are careful to note that the earlier p ≤ 37 claim may depend on GRH. These are real improvements, not repackaged results.\n\nWhat the paper does well: the reduction to Thue equations, the Galois analysis of the associated fields (including the explicit root formula and the Aff(F_p) Galois group), and the application of Laurent's linear-forms bounds are written in enough detail that the nontrivial analytic steps can actually be checked. The continued fraction argument in Section 6 is elegant, and the lower bound on the product of imaginary parts is a nice touch.\n\nThe soft spot is exactly where the stress-test note puts it: the unconditional status of Theorems 1.2, 1.4, and 1.5 depends on finite computations that are not printed. The GitHub repository exists, but the paper gives no commit pin, no full logs, and no explicit verification of the Table 1 inequalities. The tight margin around condition (5.37) for p = 919 — about 0.02 — means a rounding error in K', μ, or ρ would remove the contradiction. I do not see a circularity problem; the parameters are checked after selection. But as written, the proof is not self-contained at this step. Also, the sentence in Section 6 that describes the range for Table 1 appears to misstate it; the table itself is fine.\n\nWho is this for? Number theorists working on exponential Diophantine equations. A serious referee should engage with the paper, but should ask for the computational certificates to be fully documented and the tight inequality for p = 919 to be verified with explicit error bounds.\n\nMy recommendation: send to a good referee. The analytic framework is credible and the results are significant; the computational gap is fixable in revision.","headline":"Genuinely new partial results on a classic Diophantine equation, with an un-auditable computational core that a serious referee should push to be fully documented.","tokens_in":45178,"tokens_out":2496,"would_cite":true,"duration_ms":27581,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D61","11D41","11J86","11Y50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every odd prime $p>911$, the equation $x^2-2=y^p$ has only the trivial solution $y=-1$; any other solution would force $y>10^{1000}$.","keywords":["Lebesgue-Nagell equation","exponential Diophantine equations","trivial solutions conjecture","Thue equations","linear forms in two logarithms","continued fractions","modular method","lower bounds for integer solutions"],"falsifier":"Run the computations described in Sections 5 and 6 and check three things independently: (i) for each prime $3\\le p\\le13$ the stated Thue equations have no nontrivial solutions; (ii) for each prime $17\\le p<20000$ the listed auxiliary primes satisfy the four conditions of the proof of Theorem 5.3 and intersect to $\\{1,-1\\}$; (iii) the continued-fraction computation reproduces the Table 1 lower bounds for $919\\le p\\le1951$ and the $10^{1000}$ bound for $17\\le p\\le911$. Failure of any one of these, or a single integer solution with $x^2-2=y^p$ and $p>911$, would refute the main claim.","tokens_in":43945,"feed_emoji":"🔢","tokens_out":9653,"duration_ms":104610,"temperature":0.7,"pith_summary":"This paper attacks the Lebesgue–Nagell equation $x^2-2=y^p$ in integers, with $p$ an odd prime. A folklore conjecture says the only solutions are the trivial ones $(\\pm1,-1)$. The authors prove the conjecture unconditionally for $p\\le13$, and prove it for every prime $p>911$; in the remaining range they show any counterexample would have $y>10^{1000}$. Combined with a known modularity result, this leaves exactly 84 prime exponents between 17 and 911 that a counterexample would have to use. The proof works by converting a hypothetical solution into a solution of a Thue equation, then applying lower bounds for linear forms in two logarithms and a continued-fraction computation.","feed_headline":"For every prime p>911, x^2-2=y^p forces y=-1","feed_subtitle":"Thue reductions, logarithm bounds, and continued fractions shrink the open conjecture to 84 prime exponents.","key_machinery":"The machinery has three linked parts. First, unique factorization in $\\mathbb{Z}[\\sqrt2]$ turns a solution into $x+\\sqrt2=(1+\\sqrt2)^r(a+b\\sqrt2)^p$, whence $(a,b)$ solves the Thue equation (3.2); after the modularity step shows $r=\\pm1$, this becomes the single equation (3.3). Second, lower bounds for linear forms in two logarithms give a lower bound on $\\Lambda=\\log((x+\\sqrt2)/(x-\\sqrt2))$ that clashes with the elementary upper bound $\\log\\Lambda<1.053-\\frac p2\\log y$ once $p$ is large. Third, for the remaining small-$y$ cases, the real root $\\theta$ of the Thue polynomial has the property that $a/b$ is a continued-fraction convergent to $\\theta$; computing sufficiently many partial quotients forces $y$ above a prescribed threshold, proving the main theorems.","core_discovery":"The central discovery is a sharp quantitative reduction: a nontrivial solution $(x,y)$ to $x^2-2=y^p$ would produce a solution $(a,b)$ to a Thue equation (3.2) attached to the ring $\\mathbb{Z}[\\sqrt2]$; the paper shows, through successive applications of linear-form-in-logarithms bounds, that no such solution can exist for $p>911$, and that for $17\\le p\\le911$ any solution must satisfy $|a|,|b|,y>10^{1000}$. The proof also addresses a gap in the literature: earlier claims solving the Thue equations for $p\\le37$ may have relied on the Generalized Riemann Hypothesis, whereas this paper's computation for $p\\le13$ is unconditional.","pith_inferences":["Beyond the paper's own claims, the $y>10^{1000}$ bound implies that any search for a counterexample is hopeless in practice; only a structural or analytic argument can resolve the remaining 84 cases.","Beyond the paper's own claims, the same pipeline (Thue reduction, two-logarithm bounds, continued fractions) could be extended to smaller primes if the auxiliary-prime search or the Thue solver improves; the paper's Table 1 suggests the analytic limit is near $p=911$.","Beyond the paper's own claims, the explicit quaternary-form formula for the newform coefficient in Section 9 points to a possible modular route around the remaining computation, though the paper does not establish that route."],"forward_implications":["For every prime $p>911$, the only integer solutions to $x^2-2=y^p$ are $(\\pm1,-1)$.","If the full conjecture is false, a counterexample must occur at one of the 84 primes $17\\le p\\le911$ with $p\\equiv13,17,19,23\\pmod{24}$; conversely, checking those 84 cases would settle the conjecture.","Any counterexample to the conjecture must have $y>10^{1000}$, so brute-force search for a counterexample is hopeless.","For $919\\le p\\le1951$, any nontrivial solution would have $y$ below the explicit Table 1 thresholds, and the continued-fraction computation rules those out.","The unconditional resolution for $p\\le13$ removes the dependence on the Generalized Riemann Hypothesis that may have been present in earlier claimed ranges for small primes."],"supporting_citations":[{"why":"Supplies the factorization in $\\mathbb{Z}[\\sqrt2]$, the reduction to Thue equations, and the modularity-based proposition used to force $r=\\pm1$.","marker":"[9]"},{"why":"Provides the interpolation-determinant lower bounds for linear forms in two logarithms that drive the bounds on $p$.","marker":"[16]"},{"why":"Proves the conjecture for primes $p\\equiv1,5,7,11\\pmod{24}$, which the paper combines with its own $p>911$ result to isolate the remaining exponents.","marker":"[8]"},{"why":"Recently confirmed the older claim $p\\ge1237$ using newer logarithm bounds; the paper goes further and reaches $p>911$.","marker":"[2]"},{"why":"Provides the Frey curve and level-lowering lemmas used in the modularity step behind the proof that $r=\\pm1$.","marker":"[5]"},{"why":"Sets the classical-and-modular framework for Lebesgue–Nagell equations and the prior solution for $1\\le d\\le100$ that motivates the $d=-2$ case.","marker":"[7]"},{"why":"Supplies the continued-fraction theorem used to show $a/b$ is a convergent to $\\theta$ and to extract the lower bounds on $y$.","marker":"[14]"},{"why":"The computational algebra system whose Thue equation solver the paper invokes for the unconditional $p\\le13$ verification.","marker":"[31]"}],"fun_headline_variants":["Lebesgue-Nagell conjecture proven for all p>911","Nontrivial solutions to x^2-2=y^p force y>10^1000","For p>911, x^2-2=y^p has only y=-1 solutions","Logarithm bounds settle Lebesgue-Nagell for p>911","x^2-2=y^p: nontrivial y must exceed 10^1000"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the computer calculations behind the Thue-equation solutions, the auxiliary-prime search, and the continued-fraction bounds are correct and complete; the paper describes these computations but does not print all outputs, so the theorems depend on code a reader would have to run independently.","fun_headline_variants_meta":{"raw":{"variants":["Lebesgue-Nagell conjecture proven for all p>911","Nontrivial solutions to x^2-2=y^p force y>10^1000","For p>911, x^2-2=y^p has only y=-1 solutions","Logarithm bounds settle Lebesgue-Nagell for p>911","x^2-2=y^p: nontrivial y must exceed 10^1000"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001529,"raw_usage":{"total_tokens":6070,"prompt_tokens":844,"completion_tokens":5226,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":5113}},"tokens_in":460,"tokens_out":5226,"duration_ms":38767,"temperature":1.0,"reasoning_tokens":5113,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:48:18.431150+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the computations described in Sections 5 and 6 and check three things independently: (i) for each prime $3\\le p\\le13$ the stated Thue equations have no nontrivial solutions; (ii) for each prime $17\\le p<20000$ the listed auxiliary primes satisfy the four conditions of the proof of Theorem 5.3 and intersect to $\\{1,-1\\}$; (iii) the continued-fraction computation reproduces the Table 1 lower bounds for $919\\le p\\le1951$ and the $10^{1000}$ bound for $17\\le p\\le911$. Failure of any one of these, or a single integer solution with $x^2-2=y^p$ and $p>911$, would refute the main claim.","supporting_citations":[{"cited_title":"Number theory","cited_arxiv_id":null,"evidence_quote":"Supplies the factorization in $\\mathbb{Z}[\\sqrt2]$, the reduction to Thue equations, and the modularity-based proposition used to force $r=\\pm1$."},{"cited_title":"Linear forms in two logarithms and interpolation determinants","cited_arxiv_id":null,"evidence_quote":"Provides the interpolation-determinant lower bounds for linear forms in two logarithms that drive the bounds on $p$."},{"cited_title":"On the equations a2 − 2b6 = cp and a2 − 2 = cp","cited_arxiv_id":null,"evidence_quote":"Proves the conjecture for primes $p\\equiv1,5,7,11\\pmod{24}$, which the paper combines with its own $p>911$ result to isolate the remaining exponents."},{"cited_title":"More on consecutive multiplicatively dependent triples of integers","cited_arxiv_id":"2411.12009","evidence_quote":"Recently confirmed the older claim $p\\ge1237$ using newer logarithm bounds; the paper goes further and reaches $p>911$."},{"cited_title":"Bennett and Chris M","cited_arxiv_id":null,"evidence_quote":"Provides the Frey curve and level-lowering lemmas used in the modularity step behind the proof that $r=\\pm1$."},{"cited_title":"Classical and modular approaches to exponential Diophantine equations","cited_arxiv_id":null,"evidence_quote":"Sets the classical-and-modular framework for Lebesgue–Nagell equations and the prior solution for $1\\le d\\le100$ that motivates the $d=-2$ case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continued-fraction theorem used to show $a/b$ is a convergent to $\\theta$ and to extract the lower bounds on $y$."},{"cited_title":"Bordeaux","cited_arxiv_id":null,"evidence_quote":"The computational algebra system whose Thue equation solver the paper invokes for the unconditional $p\\le13$ verification."}],"review_version":1}