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**Zentralblatt MATH**

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

**Zbl.No: ** 715.11050

**Autor: ** Erdös, Paul; Schinzel, A.

**Title: ** On the greatest prime factor of **prod**^{x}_{k = 1}f(k). (In English)

**Source: ** Acta Arith. 55, No.2, 191-200 (1990).

**Review: ** Let f be an irreducible polynomial with integer coefficients of degree exceeding 1, denote by P(n) the largest prime divisor of n and for positive integer x let F(f,x) be the product f(1)...f(x). The first author proved [J. Lond. Math. Soc. 27, 379-384 (1952; Zbl 046.04102)] that for sufficiently large x one has P(F(f,x)) > x log^{t}x, with t = c log log log x and c = c(f) > 0, and stated the stronger bound P(F(f,x)) > x \exp (log^{d}x), (1) with a certain positive d = d(f). The authors show now

P(F(f,x)) > x \exp\exp(c(log log x)^{1/3}). with a positive absolute constant c. In an added footnote they write that the assertion (1) has been recently established by *G. Tenenbaum* [A tribute to Paul Erdös, 405-443 (1990; Zbl 713.11069)].

**Reviewer: ** W.Narkiewicz

**Classif.: ** * 11N32 Primes represented by polynomials

**Keywords: ** largest prime factor of polynomial; polynomial with integer coefficients; largest prime divisor

**Citations: ** Zbl 046.04102; Zbl 713.11069

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