Positive definite quadratic forms representing integers of the form an 2+b

Abstract

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S a ={an 2∣n≥2} or S=S a,b ={an 2+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S a -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S 1-universal quadratic forms not representing 1.