A Combinatorial Proof for Partitions of Pythagorean Triples Into Three Parts

Ivan V. Morozov

A Combinatorial Proof for Partitions of Pythagorean Triples Into Three Parts

Combinatorics 2026-07-03 v1 Number Theory

Authors: Ivan V. Morozov

Abstract

Any Pythagorean triple $\{a,b,c\}$ such that $a^{2}+b^{2}=c^{2}$ satisfies an elegant relation between its partitions into three parts, namely $p(a,3)+p(b,3)=p(c,3)$ . While this property follows from elementary analytic methods, we give the first combinatorial proof of this relation.

Keywords

Pythagorean triples partitions combinatorial proof partition function three parts

Cite

@article{arxiv.2607.03618,
  title  = {A Combinatorial Proof for Partitions of Pythagorean Triples Into Three Parts},
  author = {Ivan V. Morozov},
  journal= {arXiv preprint arXiv:2607.03618},
  year   = {2026}
}