A proof of Sylvester's conjecture
Abstract
We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime p≡4,7,8pmod9 is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class p≡8pmod9, we prove that the elliptic curve Ep:y2=x3+p2/4 has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so p is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg L-function factors as the L-function of Ep times a complementary L-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central L-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take λ-division before taking the trace, where λ=1−ω and ω is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at p. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.
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BibTeX
@misc{burungale2026proof,
title = {{A proof of Sylvester's conjecture}},
author = {Ashay Burungale and Ye Tian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14893},
eprint = {2609.14893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14893},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A proof of Sylvester's conjecture
AU - Ashay Burungale
AU - Ye Tian
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.14893
UR - https://arxiv.org/abs/2609.14893
AB - We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime p≡4,7,8pmod9 is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class p≡8pmod9, we prove that the elliptic curve E_p:y^2=x^3+p^2/4 has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so p is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg L-function factors as the L-function of E_p times a complementary L-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central L-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take λ-division before taking the trace, where λ=1−ω and ω is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at p. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.
ER -