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

  • JÖRG BRÜDERN (a1) and TREVOR D. WOOLEY (a2)

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.

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

  • JÖRG BRÜDERN (a1) and TREVOR D. WOOLEY (a2)

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