REVIEW 4 major objections 5 minor 34 references
On the Lebesgue-Nagell equation $x^2-2 = y^p$
T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read 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 $y>10^{1000}$.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Lebesgue–Nagell equation x^2 - 2 = y^p for odd primes p. Following the reduction of Bugeaud–Mignotte–Siksek, nontrivial solutions are shown to yield solutions to a family of Thue equations indexed by an integer r with |r| <= (p-1)/2. The paper proves the folklore conjecture unconditionally for p <= 13 by solving these Thue equations with PARI/GP, proves r = ±1 for 17 <= p < 20000 by a modular-method auxiliary-prime computation, and then applies Laurent's lower bounds for linear forms in two logarithms in three successive stages. The final analytic stage, combined with a parameter search summarized in Table 1 and a continued-fraction computation in Section 6, yields Theorem 1.4 (no nontrivial solutions for p > 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.
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 (4)
- [Section 5.8, Table 1, Eqs. (5.36)–(5.39)] 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 5.5, Theorem 5.3] 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 3, proof of Theorem 1.2] 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 6, Theorems 1.4 and 1.5] 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.
minor comments (5)
- [Section 5.4, proof of Theorem 5.2] In the text, 'mu = 0..508613' should read 'mu = 0.508613'.
- [Section 6, Proposition 6.4] 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).
- [Table 1] 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 5.5] 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 5.3, Lemma 5.11] 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.
Circularity Check
No significant circularity: the derivation chain uses external linear-forms bounds, standard modularity inputs, and finite computational checks that are not equivalent to the target conjecture.
full rationale
I walked the derivation chain from the factorization in Section 3 through the linear-forms arguments in Section 5 and the continued-fraction arguments in Section 6. The reduction to Thue equations is a standard factorization in Z[sqrt(2)], and the lower bounds for linear forms in logarithms are quoted from Laurent's external theorems (Propositions 5.6 and 5.7), not derived in this paper. The proof that r = ±1 follows the Bugeaud-Mignotte-Siksek modular method described in Cohen's book, not the authors' own prior work. The numerical parameters in Table 1 are proof-search choices made after the sufficient conditions (5.36) and (5.37) are derived; the paper asserts that the resulting inequalities (5.38) and (5.39) were checked for the listed rows. Even if those checks are not printed in full, checking sufficient inequalities after selecting parameters is not the same as fitting the target conclusion into the hypotheses. The continued-fraction step provides an independent lower bound on y, while Theorem 5.1 provides the complementary upper bound; the contradiction is not built in by definition. The passages that come closest to a concern are computational certificates such as 'We verified this computation in Sage and output a text file with the auxiliary primes used in the proof' and 'We ran code in Sage to compute the continued fraction expansion of theta... This proves that y exceeds the lower bounds in Table 1.' These are un-audited finite computations, which is a reproducibility and correctness risk, but not circularity. I found no load-bearing self-citation, no fitted parameter renamed as a prediction, and no claim whose proof is equivalent to its own input. Score 0.
Assumptions & free parameters
free parameters (3)
- Linear-form parameters μ, ρ (Theorem 5.2) =
μ ≈ 0.508613, ρ ≈ 7.99202
- Linear-form parameters μ, ρ (Theorem 5.17) =
μ = 1/3, ρ ≈ 22.5978
- K', L, R1, R2, μ, ρ, y0 (Table 1) =
K'=26.42-28.69, L=9-10, R1=1, R2=64-69, μ=0.57-0.59, ρ=26.3-33, y0=1050-10800
assumptions (7)
- standard math Z[√2] is a UFD and its unit group is generated by -1 and 1+√2
- domain assumption The Frey curve construction and level-lowering theorem (modularity) for the equation produce congruences with newforms of level 128
- standard math Laurent's lower bounds for linear forms in two logarithms ([16, Theorems 1,2]) are valid and correctly applied
- standard math Khinchin's continued fraction approximation theorem and the properties of convergents hold as used
- domain assumption The LMFDB integral point data for the elliptic curve Y^2 = X^3 + 98 are correct
- domain assumption The computer algebra systems PARI/GP and Sage return correct and complete outputs for the Thue solving, modular checks, and continued fraction computations
- domain assumption No Wieferich primes below 1000
Cite this review
Pith. "Pith review of On the Lebesgue-Nagell equation $x^2-2 = y^p$." pith.science (2026). https://pith.science/paper/UYEEWWWA
@misc{pith2026250712397,
author = {Pith},
title = {Pith review of: On the Lebesgue-Nagell equation $x^2-2 = y^p$},
year = {2026},
howpublished = {\url{https://pith.science/paper/UYEEWWWA}},
note = {Machine review of arXiv:2507.12397}
}
abstract
We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} in integers $x,y,p$ with $p\geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with $y=-1$. We confirm the conjecture unconditionally for $p\leq 13$, and prove the conjecture holds for $p>911$ through a careful application of lower bounds for linear forms in two logarithms. We also show that any ``nontrivial'' solution must satisfy $y > 10^{1000}$. In addition, we establish auxiliary results that may support future progress on the problem, and we revisit some prior claims in the literature.
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