The heptagonal theorem of 131: every sum of generalized heptagonal numbers that represents 1, ..., 131 is universal
Year: 2026
Type: article
Abstract: Let p_7(x) = (5x^2 - 3x)/2 be the generalized heptagonal numbers. Kamaraj, Kane and Oishi-Tomiyasu (J. Number Theory 249, 2023) proved that a sum of generalized heptagonal numbers with positive integer coefficients is universal as soon as it represents every natural number up to 3.9·10^106, and observed numerically that the true bound should be 131. This paper proves gamma_7 = 131: such a sum represents every natural number if and only if it represents 1, ..., 131. The proof is computer-assisted: the escalation argument reduces the statement to 217 quaternary and 39 quinary sums; for each quaternary sum the representation numbers are a Siegel main term plus the coefficients of a weight-2 cusp form of level up to 209 600, bounded at the bad part by Petersson's formula with the Weil bound for Kloosterman sums and by an explicit cusp-by-cusp Petersson norm bound with exact theta and Eisenstein expansions at every cusp; the finitely many candidates (about 10^11) are represented explicitly, and the 3-adic obstruction of (1,1,3,3) is removed by an exact descent. Every floating-point input is enclosed in ball arithmetic (FLINT/Arb) or covered by a rounding analysis of the code that was run. The mathematics, programs and text were produced by Claude (Anthropic) under the author's direction, audited by separate AI runs and by nine rounds of adversarial review with ChatGPT (OpenAI); no human expert has yet read the argument. The attached zip contains all programs, result files, certificates, logs, working notes, audit records and review transcripts, so that every step can be rerun with free software.
Author: Cooper, Aaron
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225585190