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On the ternary pentagonal numbers conjecture

Primarily human-written textHuman understanding: all partsmath.NT — Number Theory

Contributed by Glenn Bruda ↗

SubmitterGlenn Bruda

Version 1 / Oct 02, 2026 / CC BY 4.0

Abstract

Communicated by Guy in 1994, the ternary pentagonal numbers conjecture of Blecksmith and Selfridge asserts that every integer larger than 3306633066 is the sum of three positive pentagonal numbers. Prior to this note, it was even unknown whether every sufficiently large integer is the sum of three positive pentagonal numbers. We resolve this in the affirmative, using the landmark work of Duke and Schulze-Pillot on ternary quadratic forms to handle all sufficiently large integers nn with v3(24n+3)≤8v_3(24n+3)\leq8, and present an explicit lift to handle the nn with v3(24n+3)≥9v_3(24n+3)\geq9.

Provenance statement

Having proven that every positive integer $n$ with bounded $v_3(24n+3)$ is the sum of three positive pentagonal numbers, we asked ChatGPT 5.6 Sol to prove that every integer congruent to $(9^{\lceil k/2\rceil}-1)/8\bmod{3^k}$ (which is equivalent to $v_3(8n+1)\geq k$) is the sum of three positive pentagonal numbers for $k$ sufficiently large. ChatGPT initiated a search for, and succeeded in finding, an explicit lift from a ternary generalized pentagonal number representation $\mathbf{x}\in\ZZ^3$ of $3^{-8}(n-(3^8-1)/8)$ to a ternary positive pentagonal number representation of $n$. The proof of this lift is based on the argument provided by ChatGPT, though we have reformulated the statement and proof. All of the writing in the document is the author's own, and we take full responsibility for its mathematical correctness.

Tools used

OpenAI
ChatGPTVersion 5.6 Sol

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Version history

  1. v1Initial depositCurrentOct 02, 2026