Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 013.39003
Autor: Erdös, Paul
Title: On the representation of an integer as the sum of k k-th powers. (In English)
Source: J. London Math. Soc. 11, 133-136 (1936).
Review: Let f(m) denote the number of representations of m as the sum of k k-th powers of non-negative integers. The well-known "Hypothesis K" of Hardy and Littlewood [Math. Z. 23, 1-37 (1925)] is a conjecture that f(m) = O(m\epsilon) for every \epsilon > 0. The author proves a result in the opposite direction, namely f(m) > \exp\left{{c1 log m \over log log m}\right} for an infinity of m, where c1 is a positive number depending only on k. Of course, this does not disprove Hypothesis K. The method of proof is slightly different for odd and even k. Let us suppose that k is odd and take p1,...,pr to be consecutive primes greater than k for which (p-1,k) = 1. Let A = p1...pk, n = Ak, B/A, A = BC. Let SB denote the number of solutions of xi \leq n, xi \equiv 0 \pmod B, x1k+···+xkk\equiv 0 \pmod n, (x1k+···+xk-1k,C) = 1 in non-negative integers x1,...,xk. Then SB > {c2n^{k-1} \over log pr} and the number of solutions of xi \leq n, x1k+···+xkk\equiv 0 \pmod n is at least {c32rnk-1 \over log pr}. Hence there is an m \leq knk which is a multiple of n and for which f(m) \geq {c32r\over k log p}. Since r > {c4 log n \over log log n}, the result follows. – The proof for even k depends on the lemma: If C is a product of different primes, each of which satisfies p+k, p\equiv 3 \pmod 4, (p-1,k) = 2, then the number of solutions of xk+yk\equiv a (mod Ck) , where (a,C) = 1, is Ckprodp/c (1+p-1).
The author states that his method enables him to prove that, if a1,a2,... are integers and {1\over k1}+···+{1\over kl} = 1, then there is an infinity of m with more than \exp\left{{c log m\over log log m}\right} representation in the form a1x1k+···+alxlk (xi \geq 0).
Reviewer: Wright (Aberdeen)
Classif.: * 11P05 Waring's problem and variants
11P55 Appl. of the Hardy-Littlewood method
11D85 Representation problems of integers
Index Words: Number theory
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