Representations by x(1)(2)+2x(2)(2) + x(3)(2) + x(4)(2) + x(1)x(3) + x(1)x(4) + x(2)x(4)

Abstract

Let r(Q) (n) be the representation number of a nonnegative integer n by the quaternary quadratic form Q = x(1)(2) + 2x(2)(2) + x(3)(2) + x(4)(2) + x(1)x(3) + x(1)x(4) + x(2)x(4). We first prove the identity r(Q) (p(2)n) = r(Q) (p(2))r(Q) (n)/r(Q) (1) for any prime p different from 13 and any positive integer n prime to p, which was conjectured in Eum et al. (2011) [2]. And, we explicitly determine a concise formula for the number r(Q) (n(2)) as well for any integer n. (C) 2011 Elsevier Inc. All rights reserved.