Mixed sums of squares and triangular numbers (III)

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 = 2 m + 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 2 T m ( m > 0 ) with 2 m + 1 having no prime divisor congruent to 3 modulo 4.