Ternary quadratic forms representing a given arithmetic progression

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.

9 pages, comments are welcome!