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The representation of integers by binary additive forms

Published online by Cambridge University Press:  04 December 2007

M. A. BENNETT ,
N. P. DUMMIGAN  and
T. D. WOOLEY
M. A. BENNETT
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109-1003, e-mail: mabennet@math.lsa.umich.edu
N. P. DUMMIGAN
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109-1003, e-mail: ndumm@math.lsa.umich.edu.
T. D. WOOLEY
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109-1003, e-mail: wooley@math.lsa.umich.edu
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Abstract

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Let $a, b$ and $n$ be integers with $n ⩾ 3$. 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.

Keywords

binary formsrepresentation problemshigher degree equationsDiophantine approximationWaring‘s problem and variants
Type
Research Article
Information
Compositio Mathematica , Volume 111 , Issue 1 , March 1998 , pp. 15 - 33
Copyright
1998 Kluwer Academic Publishers