BINARY QUADRATIC FORMS WITH LARGE DISCRIMINANTS AND SUMS OF TWO SQUAREFUL NUMBERS II

Abstract

Let $\textbf{F}\,{=}\,(F_1, \ldots, F_m)$ be an $m$-tuple of primitive positive binary quadratic forms and let $U_{\textbf{F}}(x)$ be the number of integers not exceeding $x$ that can be represented simultaneously by all the forms $F_j$, $j = 1, \ldots, m$. Sharp upper and lower bounds for $U_{\textbf{F}}(x)$ are given uniformly in the discriminants of the quadratic forms.

As an application, a problem of Erdős is considered. Let $V(x)$ be the number of integers not exceeding $x$ that are representable as a sum of two squareful numbers. Then $V(x) = x(\log x)^{-\alpha+o(1)}$ with $\alpha\,{=}\,1-2^{-1/3}\,{=}\,0.206\ldots$.