We shall discuss the following results which are joint work with Stanley Xiao. Let F(x,y) be a binary form with integer coefficients, degree d (greater than 2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z2/d. Let k be an integer greater than 1 and suppose that there is no prime p such that pk divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z2/d.
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