Sylvester's 1879 conjecture fully proved after 147 years
IgorCarron · x · 2026-09-20
Mathematicians Ashay Burungale and Ye Tian posted an arXiv paper completely proving Sylvester's conjecture from 1879: every prime p ≡ 4, 7, or 8 (mod 9) is a sum of two rational cubes. Elkies handled classes 4 and 7 in 1994 and Yin recently completed the proof; this work settles the final class p ≡ 8 by showing the elliptic curve Ep: y² = x³ + p²/4 has analytic rank 1, as predicted by Birch–Swinnerton-Dyer. The proof adapts the authors' Rankin–Selberg construction, uses Chan's 3-isogeny descent to show the complementary central L-value is non-zero, and introduces a novel 'λ-division before trace' idea to bypass vanishing Hecke traces.
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