# On the number of summands in the Hilbert-Kamke problem in prime numbers

On the number of summands in the Hilbert-Kamke problem in prime numbers -- In this paper it is proved that the number of summands , which are required for the simultaneous representations of positive integers Njf1<j< n, satisfying the corresponding necessary arithmetic conditions, as sums of the jth powers of prime numbers «,· > + 1, 1 < i < , belongs to some residue class modulo Ro(n) = exp{n In + O(n)}, moreover, if > 17, then for every class of numbers Ni,...,Nn, corresponding to s modulo Ao(n), the least , which is sufficient for these representations, is determined from the inequalities 80(n) < a < sQ(n) + Ro(n) - 1, where 80(n) ~ 3an, an ~ 3n/4, --> oo, provided that the numbers NI , . . . , Nn satisfy some order conditions and are large enough. The analogous situation has arisen for simultaneous representations of NI , . . . , Nn as sums of powers of arbitrary prime numbers. 1. INTRODUCTION Let > 2, 5, NI,... , Nn be positive integers, and let J be the number of solutions of the system of equations * + ... + * = ^, j = l,...,n, (1) in prime numbers, and = N^n. For fixed > 11, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

# On the number of summands in the Hilbert-Kamke problem in prime numbers

Discrete Mathematics and Applications, Volume 3 (2) – Jan 1, 1993
12 pages

/lp/de-gruyter/on-the-number-of-summands-in-the-hilbert-kamke-problem-in-prime-oFmV7p0Br0
Publisher
de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.2.161
Publisher site
See Article on Publisher Site

### Abstract

-- In this paper it is proved that the number of summands , which are required for the simultaneous representations of positive integers Njf1<j< n, satisfying the corresponding necessary arithmetic conditions, as sums of the jth powers of prime numbers «,· > + 1, 1 < i < , belongs to some residue class modulo Ro(n) = exp{n In + O(n)}, moreover, if > 17, then for every class of numbers Ni,...,Nn, corresponding to s modulo Ao(n), the least , which is sufficient for these representations, is determined from the inequalities 80(n) < a < sQ(n) + Ro(n) - 1, where 80(n) ~ 3an, an ~ 3n/4, --> oo, provided that the numbers NI , . . . , Nn satisfy some order conditions and are large enough. The analogous situation has arisen for simultaneous representations of NI , . . . , Nn as sums of powers of arbitrary prime numbers. 1. INTRODUCTION Let > 2, 5, NI,... , Nn be positive integers, and let J be the number of solutions of the system of equations * + ... + * = ^, j = l,...,n, (1) in prime numbers, and = N^n. For fixed > 11,

### Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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