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

Let F=(F1, . . . , Fm) be an m-tuple of primitive positive binary quadratic forms and let UF(x) be the number of integers not exceeding x that can be represented simultaneously by all the forms Fj , j = 1, . . . , m. Sharp upper and lower bounds for UF(x) are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erd˝os 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)−α+o(1) with α=1 − 2−1/3 =0.206 . . . .