On sums of triangular numbers

TL;DR

This paper proves that every natural number can be expressed as a sum of four triangular numbers with specific parity conditions, confirming a conjecture by Zhi-Wei Sun.

Contribution

It establishes a new decomposition result for natural numbers into triangular numbers, resolving a conjecture by Sun.

Findings

Every natural number can be written as a sum of four triangular numbers with two even and two odd indices.

The result confirms a conjecture by Zhi-Wei Sun.

Provides a new understanding of the structure of natural numbers in terms of triangular numbers.

Abstract

We study decompositions of natural numbers into triangular summands. For instance, we prove that any natural number can be represented as a sum of four triangular numbers, two of them having even indices and the other two having odd indices. This settles a conjecture by Zhi-Wei Sun (arXiv:1503.03743v8).