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On Waring's problem: a square, four cubes and a biquadrate

Published online by Cambridge University Press:  01 September 1999

JÖRG BRÜDERN  and
TREVOR D. WOOLEY
JÖRG BRÜDERN
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Postfach 80 11 40, D–70511 Stuttgart, Germany. e-mail: [email protected]
TREVOR D. WOOLEY
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109–1109, U.S.A. e-mail: [email protected]
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Abstract

Additive representations of natural numbers by mixtures of squares, cubes and biquadrates belong to the class of more interesting special cases which form the object of attention for testing the general expectation that any sufficiently large natural number n is representable in the form

formula here

as soon as the reciprocal sum [sum ]sj=1k−1j is reasonably large. With the exception of a handful of very special problems, in the current state of knowledge the latter reciprocal sum must exceed 2, at the very least, in order that it be feasible to successfully apply the Hardy–Littlewood method to treat the corresponding additive problem. Here we remove a case from the list of those combinations of exponents which have defied treatment thus far.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 2 , September 1999 , pp. 193 - 200
Copyright
The Cambridge Philosophical Society 1999

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