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THE NUMBER OF REPRESENTATIONS BY A TERNARY SUM OF TRIANGULAR NUMBERS

Kim, Mingyu;Oh, Byeong-Kweon

  • Received : 2018.01.22
  • Accepted : 2018.05.03
  • Published : 2019.01.01

Abstract

For positive integers a, b, c, and an integer n, the number of integer solutions $(x,y,z){\in}{\mathbb{Z}}^3$ of $a{\frac{x(x-1)}{2}}+b{\frac{y(y-1)}{2}}+c{\frac{z(z-1)}{2}}=n$ is denoted by t(a, b, c; n). In this article, we prove some relations between t(a, b, c; n) and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in [6].

Keywords

representations of ternary quadratic forms;squares

Acknowledgement

Supported by : National Research Foundation of Korea