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Let F = (F1,...,Fm) be an m-tuple of primitive positive binary quadratic forms and let UF(x) be the number of integers not exceed- ing 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 Erdos 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(logx)−+o(1) with  = 1 − 2− 1 3 = 0.206...

Binary quadratic forms with large discriminants and sums of two squareful numbers