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Bull. Korean Math. Soc. 2019; 56(5): 1143-1157

Online first article August 7, 2019      Printed September 30, 2019

https://doi.org/10.4134/BKMS.b180914

Copyright © The Korean Mathematical Society.

The number of representations of a positive integer by triangular, square and decagonal numbers

Uha Isnaini, Ray Melham, Pee Choon Toh

Nanyang Technological University; University of Technology, Sydney; Nanyang Technological University

Abstract

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Let $T_aD_b(n)$ and $T_aD'_b(n)$ denote respectively the number of representations of a positive integer $n$ by $a(x^2-x)/2 +b(4y^2-3y)$ and $a(x^2-x)/2 +b(4y^2-y)$. Similarly, let $S_aD_b(n)$ and $S_aD'_b(n)$ denote respectively the number of representations of $n$ by $ax^2 +b(4y^2-3y)$ and $ax^2 +b(4y^2-y)$. In this paper, we prove 162 formulas for these functions.

Keywords: representations by binary quadratic forms

MSC numbers: Primary 11E25, 11E16

Supported by: U. Isnaini was supported by the National Institute of Education (Singapore) PhD scholarship

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