Paper Detail
Yichuan Cao, Dakai Guo, Ruichen Qiu, Ruyong Feng, Xiao-Shan Gao
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.
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@article{cao2026every,
title = {Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number},
author = {Yichuan Cao and Dakai Guo and Ruichen Qiu and Ruyong Feng and Xiao-Shan Gao},
year = {2026},
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, \textbackslash{}qquad x,y,z \textbackslash{}in \textbackslash{}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 b},
url = {https://arxiv.org/abs/2606.26035},
keywords = {math.NT, cs.SC},
eprint = {2606.26035},
archiveprefix = {arXiv},
}
{}