Ternary quadratic forms representing a given arithmetic progression

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