On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2

TL;DR

This paper establishes a new identity relating the number of representations of integers as sums of three squares, using ternary quadratic forms with specific discriminants and advanced mathematical formulas.

Contribution

It introduces a novel identity connecting representations of integers as sums of three squares with ternary quadratic forms of discriminants p^2 and 16p^2, utilizing Watson's transformations and advanced formulas.

Findings

Derived a new identity for s(p^2n)- p s(n) involving ternary quadratic forms.

Connected quadratic forms with discriminants p^2 and 16p^2 through Watson's transformations.

Applied Siegel--Weil and Smith--Minkowski formulas to prove the identity.

Abstract

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.