Quadratic forms representing large integers only

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) is an element of omega(log2(n)) boolean AND O(root n). Explicit lower and upper bounds for t(n) will be provided for some small integer n. (c) 2023 Elsevier Inc. All rights reserved.