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11 August 2020
 
  » arxiv » math.NT/0505128

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Each natural number is of the form x^2+(2y)^2+z(z+1)/2
Zhi-Wei Sun ;
Date 9 May 2005
Subject Number Theory; Combinatorics MSC-class: 11E25; 05A30; 11B65; 11D85; 11P99 | math.NT math.CO
AbstractIn this paper we investigate mixed sums of squares and triangular numbers. By means of q-series, we prove that any natural number n can be written as x^2+(2y)^2+T_z with x,y,z in Z and T_z=z(z+1)/2, this is stronger than a conjecture of Chen. Also, we can express n in any of the following forms: x^2+2y^2+T_z, x^2+2y^2+2T_z, x^2+2y^2+4T_z, x^2+4y^2+2T_z, 2x^2+2y^2+T_z, x^2+2T_y+2T_z, x^2+4T_y+T_z, x^2+4T_y+2T_z, 2x^2+T_y+T_z, 2x^2+2T_y+T_z, 2x^2+4T_y+T_z, T_x+4T_y+T_z, 2T_x+2T_y+T_z, 2T_x+4T_y+T_z. Concerning the converse we establish several theorems and make some conjectures.
Source arXiv, math.NT/0505128
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