Mixed sums of squares and triangular numbers (III)

TL;DR

This paper proves a conjecture by Sun that every positive integer can be expressed as a sum of a square, an odd square, and a triangular number, and explores related representations involving primes and quadratic forms.

Contribution

It confirms Sun's conjecture and characterizes when certain sums involving squares and triangular numbers are possible, linking these to properties of primes congruent to 3 modulo 4.

Findings

Confirmed Sun's conjecture on sums of squares and triangular numbers.

Characterized when specific sums involving odd squares and triangular numbers are impossible.

Linked the impossibility of certain representations to primes congruent to 3 modulo 4.

Abstract

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if T_m=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p^2=x^2+8(y^2+z^2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2T_m (m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.