A result similar to Lagrange's theorem

JOURNAL OF NUMBER THEORY

卷:162

Article

Sun, Zhi-Wei¹ 

[1] Nanjing Univ, Dept Math, Nanjing 210093, Jiangsu, Peoples R China

关键词: Generalized octagonal numbers;    Quadratic forms;    Representations of integers;   

DOI  :  10.1016/j.jnt.2015.10.014

来源: Elsevier

【 摘 要 】

Generalized octagonal numbers are those p8(x) = x(3x 2) with x is an element of Z. In this paper we show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for 35 triples (b, c, d) with 1 <= b <= c <= d (including (2,3,4) and (2,4,8)), we prove that any nonnegative integer can be expressed as p(8)(w) bp(8) (x) cp(8)(y) dp(8)(z) with w, x, y, z E Z. We also pose several conjectures for further research. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free