On x(ax+1) + y(by+1) + z(cz+1) and x(ax plus b) plus y(ay plus c) plus z(az plus d)

JOURNAL OF NUMBER THEORY

卷:171

Article

Sun, Zhi-Wei¹ 

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

关键词: Representations of integers;    Universal sums;    Quadratic polynomials;   

DOI  :  10.1016/j.jnt.2016.07.024

来源: Elsevier

【 摘 要 】

In this paper we first investigate for what positive integers a, b, c every nonnegative integer n can be written as x(ax + 1) + y(by + 1) + z(cz + 1) with x, y, z integers. We show that (a, b, c) can be either of the following seven triples (1, 2, 3), (1, 2, 4), (1, 2, 5), (2, 2, 4), (2, 2, 5), (2, 3, 3), (2, 3, 4), and conjecture that any triple (a, b, c) among (2, 2, 6), (2, 3, 5), (2, 3, 7), (2, 3, 8), (2, 3, 9), (2, 3,10) also has the desired property. For integers 0 <= b <= c <= d <= a with a > 2, we prove that any nonnegative integer can be written as x(ax+b)+y(ay+c)+z(az+d) with x, y, z integers, if and only if the quadruple (a, b, c, d) is among (3, 0, 1, 2), (3, 1, 1, 2), (3,1, 2, 2), (3,1, 2, 3), (4,1, 2, 3). (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free