Stewart 2008 cubic thue equations many solutions

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theorem_1_1: For every cubic binary form F with integer coefficients and nonzero discriminant there is c = c(F) > 0 such that F(x, y) = m has at least c(log m)^(1/2) solutions in integers for infinitely many positive integers m, raising Silverman's exponent 1/3 and Mahler's 1/4 to 1/2.

theorem_4_1: For integers a, b with 4a^3 + 27b^2 nonzero, the number of cube-free integers d with |d| <= T for which x^3 + axy^2 + by^3 = d with a rational point is an elliptic curve of rank at least 2 is at least C_2 T^(1/6)/(log T)^2 if ab is nonzero, C_3 T^(1/6) if a = 0, and C_4 T^(2/9) if b = 0, for all T > C_1.

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C. L. Stewart, Cubic Thue equations with many solutions, Int. Math. Res. Not. IMRN 2008, Art. ID rnn040, 11 pp.; DOI 10.1093/imrn/rnn040. Received 29 October 2007, revised 26 March 2008, accepted 28 March 2008.

The copy read for this card is the Oxford University Press publisher PDF of the eleven article pages (PDF p. $n$ is article p. $n$), with the citation line and DOI at the head of p. 1 and the publisher's download banner (naming https://academic.oup.com/imrn/article/doi/10.1093/imrn/rnn040/696448, a University of Waterloo user, 10 November 2023) on every page; it has a text layer in which the symbol $\ne$ is dropped (the page images were consulted for Theorem 4.1). Provenance: the copy was obtained in the repository's survey download of September 2026; the survey record identifies the source by the DOI 10.1093/imrn/rnn040 (https://doi.org/10.1093/imrn/rnn040), the article URL in the banner is the only URL the file names, and the download URL itself was not recorded; 143,250 bytes. The file prints "© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org." on p. 1, every other right reserved.

Read status: claims checked for Theorem 1.1 and Theorem 4.1, whose statements were read clause by clause (Theorem 1.1 in the text layer, Theorem 4.1 on the page image), and again on the page images of all eleven pages on 2026-10-08; their proofs were read but not verified; the problem page does not yet consume any statement from this source. The statements are on theorem_1_1 and theorem_4_1.

Contents

• Background (pp. 1--3): for a binary form $F$ of degree $r\ge3$ with integer coefficients and nonzero discriminant, $F(x,y)=m$ (1) has finitely many integer solutions (Thue 1909 for irreducible $F$). Lower bounds for the number of solutions: Chowla 1933, at least $c_0\log\log m$ solutions of $x^3-ky^3=m$ for infinitely many $m$; Mahler 1935 (the problem's [Ma35b]), at least $c_1(\log m)^{1/4}$ for infinitely many $m$, for any cubic $F$ of nonzero discriminant; Silverman 1983, exponent $1/3$.
• Theorem 1.1 (p. 2; deduced from Theorem 4.1 through Silverman's Theorem): for each cubic binary form $F\in\mathbb Z[x,y]$ of nonzero discriminant there is $c=c(F)>0$ with $\#\{(x,y)\in\mathbb Z^2:F(x,y)=m\}\ge c(\log m)^{1/2}$ for infinitely many positive integers $m$. Statement read clause by clause in the text layer.
• Silverman's Theorem (p. 2, quoted from Silverman 1983): if $E:F(x,y)=m_0z^3$ has a rational point and Mordell–Weil rank $r$, then $F(x,y)=m$ has at least $c_2(\log m)^{r/(r+2)}$ solutions for infinitely many positive integers $m$; so Theorem 1.1 follows once each $F$ has a twist of rank at least $2$.
• Remarks on $x^3+y^3$ (p. 3): Silverman exhibited a twist of $x^3+y^3=1$ of rank at least $3$, so $x^3+y^3=m$ has at least $c_3(\log m)^{3/5}$ solutions for infinitely many $m$; Stewart 1995 found a twist of rank at least $6$ (exponent $3/4$); Elkies and Rogers 2004 found a twist of rank $11$, so the exponent may be taken to be $11/13$. Silverman also showed that some cubic forms admit the exponent $2/3$, improved to $6/7$ by Liverance and Stewart through Quer's rank-$12$ curves, and Stewart (forthcoming, the paper's [16]) shows infinitely many inequivalent cubic forms with that exponent.
• Section 3 (p. 5): by a unimodular change of variables and scaling it suffices to treat $F(x,y)=x^3+axy^2+by^3$ with $4a^3+27b^2\ne0$.
• Theorem 4.1 (p. 7; proof in section 5, pp. 7--10; checked on the page image): for $F(x,y)=x^3+axy^2+by^3$ with integers $a,b$ and $4a^3+27b^2\ne0$, count the cube-free integers $d$ with $|d|\le T$ for which the cubic curve $F(x,y)=d$ has a rational point and, with such a point as origin, is an elliptic curve of rank at least $2$. There are positive constants $C_1,\dots,C_4$ such that for every real $T>C_1$ this count is at least $C_2T^{1/6}/(\log T)^2$ when $ab\ne0$, at least $C_3T^{1/6}$ when $a=0$, and at least $C_4T^{2/9}$ when $b=0$. The proof gives, in each case, an explicit polynomial $D(t)$ and two $\mathbb Q(t)$-points on $F(x,y)=D(t)$, maps them by the covariant isogeny (7) to $y^2=x^3+432(4a^3+27b^2)D(t)^2$, proves rank at least $2$ over $\mathbb Q(t)$ by the Stewart–Top pullback criterion (Lemma 2.2), specializes by Silverman's theorem (Lemma 2.1), and counts cube-free values by Stewart–Top (Lemmas 3.1, 3.2). MAPLE was used for many of the calculations. The print writes the form in the statement as "$F(xy)$" [sic], and the proof names the constant of the case $b=0$ $C_3$ and that of the case $a=0$ $C_4$, the reverse of the statement.

Compiled scope

The whole eleven-page paper was read in the text layer, with p. 7 also on the page image, and again on the page images of all eleven pages on 2026-10-08. The statements of Theorems 1.1 and 4.1 were checked clause by clause; the proof of Theorem 4.1 was read but its pullback computations were not verified, and the quoted results of Silverman and of Stewart and Top were not checked. Nothing here is independently reviewed.

Bears on. #829, as a lower bound for a related count: Theorem 1.1 (p. 2), deduced from Theorem 4.1 (p. 7), with $F=x^3+y^3$ gives infinitely many positive $m$ with at least $c(\log m)^{1/2}$ representations as $x^3+y^3$ in integers of either sign, and p. 3 records the exponent $11/13$ for this form. These counts allow negative coordinates, while $1_A\ast1_A(n)$ counts sums of two cubes of natural numbers, and the paper says nothing about solutions in natural numbers. The problem's source says this paper improved Mahler's bound to $1_A\ast1_A(n)\gg(\log n)^{11/13}$; that needs a further step the paper does not take. The paper proves no upper bound.

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