Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number

2026-06-24

AI Summary

This work resolves the conjecture listed as OEIS A287616 by proving that every non-negative integer can be expressed as the sum of a triangular number, a pentagonal number, and a heptagonal number. Combining natural language reasoning, symbolic computation, and AI-assisted proof generation, we establish for the first time a complete theoretical foundation demonstrating that these three classes of polygonal numbers jointly cover all non-negative integers. The proof has been formalized in Lean 4 with a high degree of automation: all components except two externally cited lemmas have been fully machine-verified, thereby substantially enhancing the reliability and reproducibility of this number-theoretic result.

Abstract

In this paper, it is proved that any nonnegative integer can be written in the following form $$ x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2, \qquad x,y,z \in \mathbb{N}. $$ This settles the conjecture recorded as OEIS A287616. All parts of the proof have been formalized in Lean 4, with the exception of two results: one externally cited theorem and one statement verified by symbolic computation. Both the natural-language proof and the Lean formalization were generated by the MechMath Agent Team developed by the authors.

Problem

• triangular number
• pentagonal number
• heptagonal number
• nonnegative integer
• additive representation

Innovation

• polygonal numbers
• formal verification
• Lean 4
• automated theorem proving
• MechMath Agent