Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  715.11050
Autor:  Erdös, Paul; Schinzel, A.
Title:  On the greatest prime factor of prodxk = 1f(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 logtx, with t = c log log log x and c = c(f) > 0, and stated the stronger bound

P(F(f,x)) > x \exp (logdx),     (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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