Recent ground-breaking work of Conway, Schneeberger, Bhargava, and Hanke shows that to determine whether a given positive quadratic form F with integer coefficients represents every positive integer (and so is universal), it is only necessary to check that F represents all the integers in an explicitly given finite set S of positive integers. The set contains either nine or twenty-nine integers depending on the parity of the coefficients of the cross-product terms in F and is otherwise independent of F. In this article we show that F represents a given positive integer n if and only if F(y1,…, yk) = n for some integers y1,…, yk satisfying |yi| ≤ (Formula Presented) i = 1,…, k, where the positive rational numbers ci are explicitly given and depend only on F. Let m be the largest integer in S (in fact m = 15 or 290). Putting these results together we have F is universal if and only if S ⊆ {F(y1,…, yk) | |yi| ≤(Formula Presented), i = 1,…, k}.

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Journal Mathematics Magazine
Williams, K.S. (2016). Bounds for the representations of integers by positive quadratic forms. Mathematics Magazine, 89(2), 122–131. doi:10.4169/math.mag.89.2.122