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**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 c_{1} 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 p_{1},...,p_{r} to be consecutive primes greater than k for which (p-1,k) = 1. Let A = p_{1}...p_{k}, n = A^{k}, B/A, A = BC. Let S_{B} denote the number of solutions of x_{i} \leq n, x_{i} \equiv 0 \pmod B, x_{1}^{k}+···+x_{k}^{k}\equiv 0 \pmod n, (x_{1}^{k}+···+x_{k-1}^{k},C) = 1 in non-negative integers x_{1},...,x_{k}. Then S_{B} > {c_{2}^{n^{k-1}} \over log p_{r}} and the number of solutions of x_{i} \leq n, x_{1}^{k}+···+x_{k}^{k}\equiv 0 \pmod n is at least {c_{3}2^{r}n^{k-1} \over log p_{r}}. Hence there is an m \leq kn^{k} which is a multiple of n and for which f(m) \geq {c_{3}2^{r}\over k log p}. Since r > {c_{4} 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 x^{k}+y^{k}\equiv a (mod C^{k}) , where (a,C) = 1, is C^{k}**prod**_{p/c} (1+p^{-1}).

The author states that his method enables him to prove that, if a_{1},a_{2},... are integers and {1\over k_{1}}+···+{1\over k_{l}} = 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 a_{1}x_{1}^{k}+···+a_{l}x_{l}^{k} (x_{i} \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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