Quadratic forms with a strong regularity property on the representations of squares

A (positive definite and non-classic integral) quadratic form is called strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any integer k ≥ 2 , there are only finitely many isometry classes of strongly s-regular quadratic forms with rank k if the minimum of the nonzero squares that are represented by them is fixed.