Papers › Ternary quadratic forms representing a given arithmetic progression
Ternary quadratic forms representing a given arithmetic progression
Tomáš Hejda, Vítězslav Kala
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A positive quadratic form is (k,ℓ)-universal if it represents all the numbers kx+ℓ where x is a non-negative integer, and almost (k,ℓ)-universal if it represents all but finitely many of them. We prove that for any k,ℓ such that k∤ℓ there exists an almost (k,ℓ)-universal diagonal ternary form. We also conjecture that there are only finitely many primes p for which a (p,ℓ)-universal diagonal ternary form exists (for any ℓ<p) and we show the results of computer experiments that speak in favor of the conjecture.
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