Tight universal quadratic forms

TL;DR

This paper investigates tight universal quadratic forms that represent all integers greater than or equal to a given number, determining minimal ranks and providing bounds for these forms.

Contribution

It characterizes all tight universal diagonal quadratic forms and establishes bounds on their minimal rank as a function of n.

Findings

All tight -universal diagonal quadratic forms are classified.

The minimal rank t(n) grows at least logarithmically and at most proportionally to the square root of n.

Explicit bounds for t(n) are provided for small n.

Abstract

For a positive integer $n$, let $\mathcal T(n)$ be the set of all integers greater than or equal to $n$. An integral quadratic form $f$ is called tight $\mathcal T(n)$-universal if the set of nonzero integers that are represented by $f$ is exactly $\mathcal T(n)$. The smallest possible rank over all tight $\mathcal T(n)$-universal quadratic forms is defined by $t(n)$. In this article, we find all tight $\mathcal T(n)$-universal diagonal quadratic forms. We also prove that $t(n) \in \Omega(\log_2(n)) \cap O(\sqrt{n})$. Explicit lower and upper bounds for $t(n)$ will be provided for some small integer $n$.