Binary quadratic forms and sums of powers of integers

TL;DR

This paper explores solutions to classical Diophantine equations using binary quadratic forms and sums of powers, providing new relations and families of solutions expressed through power sums.

Contribution

It introduces a novel method to generate infinite solutions to cubic and quadratic Diophantine equations via linear combinations of power sums.

Findings

Derived linear relations for sums of powers as solutions to Diophantine equations

Generated infinite families of solutions for cubic equations involving quadratic forms

Connected quadratic forms with sums of powers to find new solution sets

Abstract

In this methodological paper, we first review the classic cubic Diophantine equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions $q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form. Next we turn our attention to the familiar sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and express the squares $S_k^2$, $S_m^2$, and the product $S_k S_m$ as a linear combination of power sums. These expressions, along with the above quadratic-form solution for the cubic equation, allows one to generate an infinite number of relations of the form $Q_1^3 + Q_2^3 + Q_3^3 = Q_4^3$, with each $Q_i$ being a linear combination of power sums. Also, we briefly consider the quadratic Diophantine equations $a^2 + b^2 + c^2 = d^2$ and $a^2 + b^2 = c^2$, and give a family of corresponding solutions $Q_1^2 + Q_2^2 + Q_3^2 = Q_4^2$ and $Q_1^2 + Q_2^2 = Q_3^2$ in terms of sums of powers of integers.