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**Zentralblatt MATH**

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

**Zbl.No: ** 401.10003

**Autor: ** Erdös, Paul; Szekeres, G.

**Title: ** Some number theoretic problems on binomial coefficients. (In English)

**Source: ** Aust. Math. Soc. Gaz. 5, 97-99 (1978).

**Review: ** In this paper some problems which are simple to state but probably difficult to solve are posed concerning binomial coefficients. Let P(m,n) denote the greatest prime factor of (m,n). Then the authors conjecture that if 1 \leq j \leq n/2 then P(\binom ni, \binom nj] \geq i with equality holding only in a few special cases (several of which are given). If f(n) = **max**_{1 < j \leq n/2}(n,[\binom nj)) it is not difficult to show that f(n) \geq p(n) is the smallest prime factor of n, and that if n is not a prime power then f(n) \leq n/P(n) where P(n) is the greatest prime power which divides n. The authors remark that it would be of interest to characterize those n for which f(n) = n/P(n). (For example, f(30) = 6.) They also mention that it seems likely that f(n) > \sqrt n for infinitely many n.

**Reviewer: ** P.Hagis

**Classif.: ** * 11A05 Multiplicative structure of the integers

05A10 Combinatorial functions

11A41 Elemementary prime number theory

00A07 Problem books

**Keywords: ** binomial coefficients; prime factors

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