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**Zentralblatt MATH**

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

**Zbl.No: ** 419.10040

**Autor: ** Erdös, Paul; Sárközy, András

**Title: ** On the prime factors of \binom{n}{k} and of consecutive integers. (In English)

**Source: ** Util. Math. 16, 197-215 (1979).

**Review: ** Let m_{k} denote the smallest integer m such that \binom mk has more than k distinct prime factors. It was shown by *P.Erdös, H.Gupta* and *S.P.Khare* [Utilitas Math. 10, 51-60 (1976; Zbl 339.10006)] that m_{k} > C k^{2} log k. Using deep results on the distribution of primes the present authors show that log k can replaced by (log k)^{4/3}(log log k)^{-4/3}(log log log k)^{-1/3}. Also let n_{k} denote the smallest integer n such that the numbers n+1, ..., n+k all have a prime factor exceeding k. It is shown by elementary means that, for sufficiently large k, n_{k} > \frac 1{16}k^{5/2}. This bounded is probably nowhere near the best possible.

**Reviewer: ** I.Anderson

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

11A41 Elemementary prime number theory

05A10 Combinatorial functions

**Keywords: ** binomial coefficient; consecutive integers; distinct prime factors

**Citations: ** Zbl.339.10006

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag