Sharpening effective upper bounds for integer solutions of the Mordell equation $y^2 = x^3 + k$

For each nonzero integer $k$, the Mordell equation $y^2 = x^3 + k$ has only finitely many integer solutions, and Baker's 1968 paper converted this qualitative finiteness into the first fully explicit bound $\max(|x|,|y|) < \exp\{(10^{10}|k|)^{10^4}\}$, with numerical constants far from what the method should deliver. This statement concerns the research program launched by that gap: Stark's effective bound $\log\max\{|x|,|y|\} \ll_\epsilon |k|^{1+\epsilon}$ (1973), the explicit-constant refinements of Sprindžuk, Hajdu-Herendi, and Juricevic, von Känel's modularity-based bound (2014), the complete computational resolution of all $0 < |k| \le 10^7$ by Bennett-Ghadermarzi (2015), and Pasten's power-saving bound $\log\max\{|x|,|y|\} \ll |k|^{1/2}(\log(2|k|))^4$ (August 2026), which is the first improvement of the $|k|$-exponent in over fifty years. The surviving open core is precise: prove a new effective upper bound on $\log\max\{|x|,|y|\}$, uniform over all nonzero $k$, with asymptotically strictly smaller growth than $|k|^{1/2}(\log(2|k|))^4$ (in particular any reduction of the exponent $1/2$), thereby taking the next rigorous step from the current record toward the $O_\epsilon(\log|k|)$ growth predicted by Hall's conjecture.

Background

The Mordell equation is the Diophantine equation $y^2 = x^3 + k$ in integer unknowns $x, y$, for a fixed nonzero integer parameter $k$. Its rational solutions form an elliptic curve (a smooth projective curve of genus one with a chosen origin, given here in short Weierstrass form), and its integer solutions are called the integral points of that curve. Mordell proved in the 1920s, building on Thue's work, that for each fixed $k$ the equation has only finitely many integer solutions; a Thue equation is an equation $F(x,y) = m$ where $F$ is a homogeneous irreducible binary form (a homogeneous polynomial in two variables) of degree at least $3$, and the classical reduction associates to a solution of $y^2 = x^3 + k$ the binary cubic form $X^3 - 3xXY^2 - 2yY^3$ of discriminant $-108k$ and certain cubic Thue equations attached to it. Mordell's finiteness argument, like Siegel's more general theorem on integral points, is ineffective: it proves that a finite search would terminate without yielding any computable bound on where the search must end.

Effectivity came from Baker's theory of linear forms in logarithms, which gives explicit lower bounds for nonzero expressions $b_1\log\alpha_1 + \dots + b_n\log\alpha_n$ with algebraic $\alpha_i$ and integer $b_i$. The source paper of this problem (Baker 1968) made the earlier effective arguments fully explicit and proved its Theorem 1: every integer solution of $y^2 = x^3 + k$ satisfies

$$\max(|x|,|y|) < \exp\left\{\left(10^{10}|k|\right)^{10^4}\right\}.$$

This closed the explicitness gap but with grotesquely non-optimal constants, obtained from many deliberately conservative numerical simplifications in the chain of estimates; the size of the bound was recognized as far from what the method should deliver, and improving it became the natural continuation.

The subsequent improvement program has two prongs. On the asymptotic prong, Stark (1973) proved that $\log\max\{|x|,|y|\} \ll_\epsilon |k|^{1+\epsilon}$ with an effective constant depending only on $\epsilon$ (here $\ll$ means inequality up to a constant factor, and an effective constant is one that is in principle computable). This shape was refined but not fundamentally improved for over fifty years: Sprindžuk's monograph systematized the effective theory of classical equations; Hajdu and Herendi (1998) gave the best fully explicit bounds for general elliptic equations, of roughly $|k|^2(\log|k|)^{10}$-type growth for Mordell equations, and Juricevic (2008) improved the shape to roughly $|k|(\log|k|)^4$-type growth in the logarithm of the solution size for Mordell equations over $\mathbb{Z}$, with explicitly written but enormous constants; and von Känel (2014) found a fundamentally different, modularity-based proof (using the Shimura-Taniyama conjecture, now the modularity theorem asserting that every elliptic curve over the rationals arises from a modular form, together with Faltings heights of elliptic curves, where the height is a logarithmic measure of arithmetic complexity) yielding the explicit bound $h(x), h(y) \le h(k) + 4a_S(\log a_S)^2$, with $h$ the logarithmic Weil height (for a rational number in lowest terms, the logarithm of the maximum of the absolute values of its numerator and denominator) and $a_S$ a truncated version of $k$ bounded by a constant times $|k|$, which improved the previous explicit records and gave a new proof of Baker's effective finiteness not relying on transcendence techniques. On the practical prong, the equation has been completely solved for ever larger ranges of $k$: Gebel, Pethő and Zimmer (1998) determined all solutions for $|k| < 10^4$ using linear forms in elliptic logarithms (elliptic analogues of Baker's logarithmic estimates, adapted to points on a fixed elliptic curve); Bennett and Ghadermarzi (2015) reached all $0 < |k| \le 10^7$ through the classical reduction to cubic Thue equations combined with lower bounds for linear forms in logarithms and lattice-basis reduction (the LLL algorithm for finding short vectors in a lattice); and von Känel and Matschke (2023), followed by a Lean 3 formalization of class-group-based resolutions by Baanen, Best, Coppola and Dahmen (2023), developed modularity-based algorithms covering large classes of Mordell equations.

In August 2026 Pasten proved a new effective bound for cubic Thue-Mahler equations (Thue equations whose right-hand side is allowed to vary over integers with prime factors in a fixed finite set) with power-saving dependence on the regulator (a measure of the size of a unit lattice of a number field), and deduced that all integer solutions of $y^2 = x^3 + k$ satisfy $\log\max\{|x|,|y|\} \ll |k|^{1/2}(\log(2|k|))^4$ with an effective absolute constant: the first power saving in the $|k|$-aspect, without restrictions on $k$, in more than fifty years. This is the current record. For calibration, Hall's conjecture on the separation between squares and cubes asserts $|x^3 - y^2| \gg_\epsilon x^{1/2-\epsilon}$ whenever $x^3 \ne y^2$ (supported by extensive computation, notably Elkies's lattice-reduction searches); applied to $y^2 = x^3 + k$ it predicts $x \ll_\epsilon |k|^{2+\epsilon}$, hence $\max\{|x|,|y|\} \ll_\epsilon |k|^{3+\epsilon}$, that is, $\log\max\{|x|,|y|\} \ll_\epsilon \log|k|$ — an unconditional polynomial bound on the largest solution in terms of $|k|$. The gap between the proven $|k|^{1/2}(\log|k|)^4$ and the conjectural $\log|k|$ is the open territory that the source paper's question has become.

Problem Statement

Let $k$ be a nonzero integer and let $(x,y) \in \mathbb{Z}^2$ range over the integer solutions of the Mordell equation $y^2 = x^3 + k$. The problem is to sharpen the effective quantitative control of the solution sizes in terms of $|k|$ beyond the current record: prove, with a complete and effective argument, a new upper bound

$$\log\max\{|x|,|y|\} \le C\,F(|k|)$$

valid for every nonzero integer $k$ and every integer solution, where $F$ is an explicitly given growth function and $C$ is an effectively computable (ideally fully explicit) constant, such that $F(t) = o\left(t^{1/2}(\log 2t)^4\right)$ as $t \to \infty$; in particular, reducing the exponent $1/2$ of $|k|$ (a further power saving) is the strong form of the requested improvement. A further goal of this program is an explicit-constant refinement at the same record growth shape $|k|^{1/2}(\log(2|k|))^4$ — the record bound of Pasten (2026) has only an effective and absolute implicit constant — improving the explicit bound provided by Baker. The bound must be uniform in $k$: no restriction on the sign, size, or prime factorization of $k$ (bounds exploiting special factorizations, such as von Känel's truncation by $\prod_{p|k}p^{\min(2,v_p(k))}$, already exist and do not by themselves answer the question), and no unhandled exceptional set of $k$. No particular method is required. For calibration, the conjectural endpoint is the Hall-conjecture/abc-strength bound $\log\max\{|x|,|y|\} \ll_\epsilon \log|k|$ (implied by Hall's prediction $x \ll_\epsilon |k|^{2+\epsilon}$, since $y^2 = x^3 + k$ then gives $\max\{|x|,|y|\} \ll_\epsilon |k|^{3+\epsilon}$); the problem asks for the next rigorous step from the current record $\ll |k|^{1/2}(\log(2|k|))^4$ toward that endpoint.

References

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