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The Waring–Goldbach problem for cubes with an almost prime
Author(s) -
Kawada Koichi,
Zhao Lilu
Publication year - 2019
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms.12244
Subject(s) - mathematics , congruence (geometry) , combinatorics , goldbach's conjecture , prime (order theory) , product (mathematics) , prime number , arithmetic , discrete mathematics , geometry
We show that every sufficiently large even integer can be written as the sum of eight cubes, seven of which are cubes of primes, and the remaining one is that of the product of two primes. We also prove that almost all natural numbers n satisfying some necessary congruence conditions may be written in the form n = p 1 3 + p 2 3 + p 3 3 + x 3 , where the p j are primes and x is the product of two primes. These conclusions should be compared with results in Kawada ( Arch. Math . 69 (1997) 13–19) and Brüdern and Kawada ( Glasg. Math. J . 51 (2009) 703–712).