Conjectures of Sun about sums of polygonal numbers

TL;DR

This paper proves that certain sums of generalized m-gonal numbers represent all positive integers beyond a specific bound, confirming Sun's conjecture for large integers.

Contribution

It verifies Sun's conjecture by establishing a finite bound up to which all positive integers are represented by these sums.

Findings

Sums of generalized m-gonal numbers represent all positive integers beyond a bound C_m.

Verification of Sun's conjecture for sufficiently large positive integers.

Explicit bounds C_m are identified for the representation property.

Abstract

In this paper, we show that certain sums of generalized $m$-gonal numbers represent every positive integer if and only if they represent every positive integer up to an explicit bound $C_m$, verifying a conjecture of Sun for sufficiently large positive integers.