Published in

Elsevier, Journal of Number Theory, 4(129), p. 964-969, 2009

DOI: 10.1016/j.jnt.2008.10.002

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Mixed sums of squares and triangular numbers (III)

Journal article published in 2009 by Byeong-Kweon Oh, Zhi-Wei Sun
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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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 Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p2=x2+8(y2+z2) 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.