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) limsupn ––> oo P(n!+1)/n > 2+\delta where \delta is an effectively computable positive constant. The authors also prove: Theorem. Let pn denote the n-th prime number. Then for infinitely many integers n(> 0), P(p1 ... pn+1) > pn+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 prodp \leq np = xm-ym and prodp \leq np = xm+ym 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

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