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A Note on Waring's Problem

A Note on Waring's Problem R. J. COOK G(k) is defined to be the least positive integer such that all sufficiently large positive integers can be expressed as the sum of at most G(/c) /cth powers of positive integers. Previously the best results known for G(9) and G(10), due to Narasimhamurti [5], were G(9) < 99 and G(10) < 122. For further results and references see Ellison [3]. THEOREM. G(9) ^ 96 and G(10) < 121. (1) These improvements were obtained by using a computer to repeatedly iterate the method of Davenport described below. Let N(s, k ; n) denote the number of positive integers not exceeding n which are representable as the sum of s fcth powers of positive integers. Suppose that a(s, k) is such that for any e > 0 k) E N(s,k;n)>rf^ - for n>n (s k,e). (2) o > k 1 We put K = 2 ~ and choose the integer t so that >k (3) and (+l) , (4) where T(k) is defined in [4]. Then, by an application of the Hardy-Littlevvood method, we have G(k) From Table 1 of Hardy and Littlewood [4] we see that T(9) = 13 and T(10) = 12. For k = 9, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/5.1.11
Publisher site
See Article on Publisher Site

Abstract

R. J. COOK G(k) is defined to be the least positive integer such that all sufficiently large positive integers can be expressed as the sum of at most G(/c) /cth powers of positive integers. Previously the best results known for G(9) and G(10), due to Narasimhamurti [5], were G(9) < 99 and G(10) < 122. For further results and references see Ellison [3]. THEOREM. G(9) ^ 96 and G(10) < 121. (1) These improvements were obtained by using a computer to repeatedly iterate the method of Davenport described below. Let N(s, k ; n) denote the number of positive integers not exceeding n which are representable as the sum of s fcth powers of positive integers. Suppose that a(s, k) is such that for any e > 0 k) E N(s,k;n)>rf^ - for n>n (s k,e). (2) o > k 1 We put K = 2 ~ and choose the integer t so that >k (3) and (+l) , (4) where T(k) is defined in [4]. Then, by an application of the Hardy-Littlevvood method, we have G(k) From Table 1 of Hardy and Littlewood [4] we see that T(9) = 13 and T(10) = 12. For k = 9,

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1973

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