On the representations of integers by the sextenary quadratic form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2 and 7-cores

TL;DR

This paper derives an explicit formula for counting integer representations by a specific sextenary quadratic form and explores inequalities relating these counts to 7-core partitions, revealing new connections in number theory.

Contribution

It provides a novel explicit formula for representations by a sextenary quadratic form and establishes inequalities linking these counts to 7-core partitions.

Findings

Explicit formula for the number of representations by the sextenary form

Inequalities relating representation counts to 7-core partitions

Connections between quadratic forms and partition theory

Abstract

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to 0,2,6,16. Here a_7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of n+2 by the sextenary form (x ^2+ y ^2+z ^2+ 7s ^2 + 7t ^2+ 7u^2)/8 with x,y,z,s,t and u being odd.