Quadratic forms representing large integers only

Abstract

For a positive integer n, let T(n) be the set of all integers greater than or equal to n. An integral quadratic form f is called tight T(n)-universal if the set of nonzero integers that are represented by f is exactly T(n). Let t(n) be the smallest possible rank over all tight T(n)-universal quadratic forms. In this article, we find all tight T(n)-universal diagonal quadratic forms. We also prove that t(n)∈Ω(log2⁡(n))∩O(n). Explicit lower and upper bounds for t(n) will be provided for some small integer n.