##
**Zentralblatt MATH**

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

**Zbl.No: ** 332.10028

**Autor: ** Erdös, Paul; Stewart, C.L.

**Title: ** On the greatest and least prime factors of nÜ+1. (In English)

**Source: ** J. London Math. Soc., II. Ser. 13, 513-519 (1976).

**Review: ** The subject of largest prime factors of special sequences of positive integers offers many interessting and challenging problems which are very simple to state. The introduction to the paper gives a short account of these. in the introduction P(f(1) ... f(x)) < Cx log x is an over sight; the inequality should be in the opposite direction. The authors prove (denoting by P(m) the largest prime factor of m): Theorem: (i) For all positive integers n, P(n!+1) > n+(1-o(1) log n/ log log n. (ii) Let \epsilon (n) be any positive function of n which tends to zero as n tends to infinity. Then for almost all integers n, P(n!+1) > n+\epsilon (n)n^{ ½}. (iii) **limsup**_{n ––> oo} P(n!+1)/n > 2+\delta where \delta is an effectively computable positive constant. The authors also prove: Theorem. Let p_{n} denote the n-th prime number. Then for infinitely many integers n(> 0), P(p_{1} ... p_{n}+1) > p_{n+k} where k > c log n/ log log n for some positive absolute constant c. In proving the latter theorem the authors also establish: Theorem. The equations **prod**_{p \leq n}p = x^{m}-y^{m} and **prod**_{p \leq n}p = x^{m}+y^{m} have no solutions in positive integers x,y,n(> 2) and m(> 1).

**Reviewer: ** K.Ramachandra

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

11N37 Asymptotic results on arithmetic functions

11A41 Elemementary prime number theory

11D41 Higher degree diophantine equations

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag