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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,
Bulletin of the London Mathematical Society – Wiley
Published: Mar 1, 1973
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