Mixed sums of squares and triangular numbers (II)

Song Guo; Hao Pan; Zhi-Wei Sun

paper

Mixed sums of squares and triangular numbers (II)

arXiv:math/0505187

Abstract

For an integer $x$ let $t_{x}$ denote the triangular number $x (x + 1) /2$ . Following a recent work of Z. W. Sun, we show that every natural number can be written in any of the following forms with $x, y, z \in Z$ : $x^{2} + 3 y^{2} + t_{z}, x^{2} + 3 t_{y} + t_{z}, x^{2} + 6 t_{y} + t_{z}, 3 x^{2} + 2 t_{y} + t_{z}, 4 x^{2} + 2 t_{y} + t_{z} .$ This confirms a conjecture of Sun.