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Additive representation in thin sequences, VI: representing primes, and related problems

Published online by Cambridge University Press:  26 February 2003

J. Brüdern
Mathematisches Institut A, Universität Stuttgart, D-70511 Stuttgart, Germany e-mail:
K. Kawada
Department of Mathematics, Faculty of Education, Iwate University, Morioka 020-8550, Japan e-mail:
T. D. Wooley
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, U.S.A. e-mail:
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We discuss the representation of primes, almost-primes, and related arithmetic sequences as sums of kth powers of natural numbers. In particular, we show that on GRH, there are infinitely many primes represented as the sum of 2\lceil 4k/3\rceil positive integral kth powers, and we prove unconditionally that infinitely many P_2-numbers are the sum of 2k+1 positive integral kth powers. The sieve methods required to establish the latter conclusion demand that we investigate the distribution of sums of kth powers in arithmetic progressions, and our conclusions here may be of independent interest.

Research Article
2002 Glasgow Mathematical Journal Trust