The Journal of the Indian Mathematical Society

Abstract

Hardy and Wright derived the probability that two integers should be relatively prime from the asymptotic formula for

Σ Φ(n).

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Abstract

In a manuscript, L. K. Hua has proved, among other things, that

G(5) ≤ 24, G(6) ≤ 37 and G(7) ≤ 53,

these being improvements on my results. Before the receipt of Hua's manuscript, I had pointed out that G(5) ≤ 24. The main lemma of Hua's argument is aimed at improving the lower limits for H_s,k(N), the number of integers less than N and expressible as the sum of s positive kth powers.

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Abstract

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Abstract

Consider the following way of representation of any positive integer x in the form

x = x_l^k+x^k₂+...+x^k_s,                                                   (1)

where x₁, x₂, ..., x_s are integers given by

(x₁+1)^k > x ≥ x₁^k

(x₂+1)^k > x-x₁^k ≥ x₂^k                                                 (2)

................

.................

the process terminating with

(x_s+1)^k > x-x₁^k-x^k₂-........-x^k_s-1 = x^k_s.

It can easily be seen that there is one and only one way of representation of an integer x in this manner. The number of kth powers required in the representation of x in the above manner is clearly a function of x and k, and hence can be denoted by S_k(x).

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