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**Zentralblatt MATH**

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

**Zbl.No: ** 525.10023

**Autor: ** Erdös, Paul; Pomerance, Carl

**Title: ** An analogue of Grimm's problem of finding distinct prime factors of consecutive integers. (In English)

**Source: ** Util. Math. 24, 45-65 (1983).

**Review: ** For n natural number, let f(n) denote the largest integer such that for each m in **{**n+1,...,n+f(n)**}** there is a divisor d_{m} of m with 1 < d_{m} < m and such that the d_{m}'s are all different. The authors prove that for every \epsilon > 0, n^{ ½} << f(n) << n^{1/12+\epsilon}. The lower bound is then strengthened to (1) **liminf** f(n)^{ ½} \geq 4. Moreover, equality holds in (1) if and only if there are infinitely many twin primes. Several other related results are also given.

**Reviewer: ** S.W.Graham

**Classif.: ** * 11N05 Distribution of primes

**Keywords: ** distinct prime factors of consecutive integers; Grimm conjecture

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