Breakthrough Tracker record

A preprint claims the remaining case of Sylvester’s rational-cubes conjecture

Burungale and Tian claim that every prime congruent to 4, 7 or 8 modulo 9 is a sum of two rational cubes, closing the previously open residue-8 case through an analytic-rank-one result for an associated elliptic curve.

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Stable ID: math-sylvester-rational-cubes-conjecture-2026
Revision: math-sylvester-rational-cubes-conjecture-2026.v1
Field: Mathematics · Number theory, elliptic curves and Diophantine equations
Evidence: Tier 1 · Peer reviewed: No
Record state: Provisional · Provisional claimed proof
Last checked: 2026-10-02

AI role

No substantive AI role was disclosed in the inspected manuscript.

Record details

Problem or result: Sylvester’s rational-cubes conjecture for primes in the 4, 7 and 8 residue classes modulo 9
Authors: Ashay A. Burungale and Ye Tian
Institutions: University of Texas at Austin; Morningside Center of Mathematics, Chinese Academy of Sciences
Result date: Submitted September 14 and revised September 15, 2026

Why it matters

The result would close the last residue class in a rational-cube problem rooted in Sylvester’s 1879 work and connects it with Heegner points and Gross–Zagier methods.

Limits

The long technical preprint is not peer reviewed. No formalization or independent proof-level assessment was located, and its historical formulation is modern rather than a verbatim 1879 statement.

Sources

Primary: Burungale and Tian, Sylvester’s rational-cubes conjecture

Correction and revision history

2026-10-02 — Added after primary-source, scope, status, AI-role and limitation review.

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