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Some Diophantine equations with almost all solutions trivial

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

George Greaves
George Greaves
Affiliation:
School of Mathematics, University of Wales, Cardiff, 23 Senghennydd Road, P.O. Box 926, Cardiff CF2 4YH
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In this paper we investigate the solutions in integers x, y, z, X, Y, Z of the system

where P is a real number that may be taken to be arbitrarily large, and the (fixed) integer exponent h satisfies h≥4. The system has 6P3 + O(P2) “trivial” solutions in which x, y, z are a permutation of X, Y, Z. Our result implies that the number of non-trivial solutions is at most o(P3), so that the total number of solutions is asymptotic to 6P3.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation

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Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 14 - 36
Copyright
Copyright © University College London 1997

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