THE REPRESENTATION OF INTEGERS BY BINARY ADDITIVE FORMS

Let a, b and n be integers with n greater than or equal to 3. We show that, in the sense of natural density, almost all integers represented by the binary form ax(n) - by(n) are thus represented essentially uni quely. By exploiting this conclusion, we derive an asymptotic formula for the total number of integers represented by such a form. These con clusions augment earlier work of Hooley concerning binary cubic and qu artic forms, and generalise or sharpen work of Hooley, Greaves, and Sk inner and Wooley concerning sums and differences of two nth powers.