Kaplansky's ternary quadratic form

James Kelley

doi:10.1155/s0161171201005294

Abstract

This paper proves that ifNis a nonnegative eligible integer, coprime to 7, which is not of the formx2+y2+7z2, thenNis square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integernp2by two quadratic forms in the same genus, thepth coefficient of anL-function of a suitable elliptic curve, and the class number formula prove the theorem for large primes, leaving 3 cases which are easily numerically verified.

Highlights

A quadratic form is a homogeneous polynomial of degree 2 in several variables
We say that a quadratic form in n variables, f represents t if there exists an x ∈ Zn such that t = x Ax
Kaplansky showed that all eligible integers divisible by 4 or 9 can be represented by φ1, which brings up the question of whether this pattern holds true for other perfect squares

Summary

Introduction

A quadratic form is a homogeneous polynomial of degree 2 in several variables. It is useful to consider the symmetric n-by-n matrix, A, associated with a quadratic form in n variables, that is, f (x1, x2, . . . , xn) = x Ax, where x = (x1, x2, . . . , xn). We say that a quadratic form in n variables, f represents t if there exists an x ∈ Zn such that t = x Ax. We call x primitive if gcd(x1, x2, . We say a nonnegative integer t is eligible with respect to a form f if there are no modularity conditions preventing f from representing n.

Results

Conclusion