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 uniquely. By exploiting this conclusion, we derive an asymptotic formula for the total number of integers represented by such a form. These conclusions augment earlier work of Hooley concerning binary cubic and quartic forms, and generalise or sharpen work of Hooley, Greaves, and Skinner and Wooley concerning sums and differences of two nth powers.
|Number of pages||19|
|Publication status||Published - Mar 1998|
- binary forms
- representation problems
- higher degree equations
- Diophantine approximation
- Waring's problem and variants