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 dm of m with 1 < dm < m and such that the dm's are all different. The authors prove that for every \epsilon > 0,

n ½ << f(n) << n1/12+\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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