M ay 2 00 8 Preprint , arXiv : 0804 . 3750 MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS ( III )

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