Mixed sums of squares and triangular numbers (II)

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 $x,y,z\in\Z$: $$x^2+3y^2+t_z, x^2+3t_y+t_z, x^2+6t_y+t_z, 3x^2+2t_y+t_z, 4x^2+2t_y+t_z.$$ This confirms a conjecture of Sun.

Related Organizations

Nanjing University
China (People's Republic of)
Shanghai Jiao Tong University
China (People's Republic of)

Keywords

11E25; 11D85; 11P99, Representation problems, Mathematics - Number Theory, FOS: Mathematics, Mathematics - Combinatorics, 11E25, Number Theory (math.NT), Combinatorics (math.CO), Sums of squares and representations by other particular quadratic forms, 11D85, 11P99

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