Everything You Wanted To Know About ax²+by²+cz²+dt² But Were Afraid To Ask

Integral Quadratic Forms Avoiding Arithmetic Progressions

“…In a recent paper [11], Williams posed and answered a variety of interesting questions regarding the integers represented by diagonal positive definite quaternary integral quadratic forms. Among them are several which relate to the avoidance of arithmetic progressions by the represented value set of a quadratic form of this type.…”

“…What about two arithmetic progressions? Both of these questions are answered in the affirmative in [11] by producing specific forms having these properties. The proofs given there are elementary and are based on the classical theorem of Lagrange on sums of four squares.…”