Integral Quadratic Forms Avoiding Arithmetic Progressions

TL;DR

This paper constructs specific positive definite diagonal quadratic forms that omit integers in exactly k arithmetic progressions, providing a comprehensive classification for the case k=1.

Contribution

It demonstrates the existence of such quadratic forms for all positive integers k and fully characterizes the forms when k=1.

Findings

Existence of quadratic forms avoiding integers in k arithmetic progressions

Complete classification of forms for k=1

Forms are positive definite diagonal quaternary

Abstract

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1, all forms with this property are determined.