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 mk 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 mk > C k2 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 nk 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, nk > \frac 1{16}k5/2. 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

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