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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 012.01201

**Autor: ** Erdös, Pál; Turán, Pál

**Title: ** Ein zahlentheoretischer Satz. (A number-theoretical theorem.) (In German)

**Source: ** Mitteil. Forsch.-Inst. Math. Mech. Univ. Tomsk 1, 101-103 (1935).

**Review: ** Let a be a fixed integer, and let l(k) be defined (for any k prime to a) as the least positive integer for which a^{l(k)\equiv} 1 (mod k). Generalising a result of *N.P.Romanoff* (Zbl 009.00801), the authors prove here that **sum**_{k} {1 \over kl(k)^{\epsilon}} converges for every \epsilon > 0. It suffices to prove that **sum** ^{1}/_{k} extended over those k for which l(k) < (log k)^{2 \over \epsilon} converges. For this it suffices that the number of divisors \leq n of (a-1) (a^{2}-1)...(a^{N}-1) should be O(n/ log ^{2} n), where N = **[**(log n)^{2 \over \epsilon} **]**. This is proved by estimating the number of prime factors, and considering separately those divisors with more than \sqrt{log n} different prime factors and those with less.

**Reviewer: ** Davenport (Cambridge)

**Classif.: ** * 11B25 Arithmetic progressions

11N13 Primes in progressions

**Index Words: ** Number theory

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