Representations of Arithmetic Progressions by Positive Definite Quadratic Forms

Abstract

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

Ternary universal sums of generalized polygonal numbers

“…(a, b, c) = (1, 1, 1), (1,1,2), (1,1,4), (1,2,1), (1,2,2), (1,2,3), (1,2,4), (2,2,1), (2,4,1), (2,5,1), (1,3,1), (1,4,1), (1,4,2), (1,6,1), (1,8,1), and the ternary sum ap 3 (x)+bp 4 (y)+cp 4 (z) is universal if and only if (a, b, c) is one of the following triples:…”

Diagonal odd-regular ternary quadratic forms

“…In [13], the notion of S d,a -regularity is introduced. A positive definite integral quadratic form f is called S d,a -regular if the following two conditions hold;…”