2 00 8 Preprint ( April 23 , 2008 ) MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS ( III )

In this paper we prove 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 mod 4 if and only if Tm = m(m + 1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p = x + 8(y + z) 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 2Tm with 2m+1 having no prime divisor congruent to 3 mod 4.