# Ternary quadratic forms representing a given arithmetic progression

## TL;DR

This paper investigates quadratic forms that represent all or almost all numbers in specific arithmetic progressions, proving existence results and conjecturing finiteness of certain universal forms, supported by computational evidence.

## Contribution

It establishes the existence of almost universal diagonal ternary quadratic forms for any suitable parameters and proposes a conjecture on the finiteness of primes with universal forms, supported by experiments.

## Key findings

- Existence of almost (k, l)-universal diagonal ternary forms for all k, l with k not dividing l.
- Conjecture that only finitely many primes p admit a (p, l)-universal form.
- Computational experiments support the conjecture.

## Abstract

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$ such that $k\nmid\ell$ there exists an almost $(k,\ell)$-universal diagonal ternary form. We also conjecture that there are only finitely many primes $p$ for which a $(p,\ell)$-universal diagonal ternary form exists (for any $\ell<p$) and we show the results of computer experiments that speak in favor of the conjecture.