Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  337.10001
Autor:  Erdös, Paul
Title:  Problems and results on number theoretic properties of consecutive integers and related questions. (In English)
Source:  Proc. 5th Manitoba Conf. numer. Math., Winnipeg 1975, 25-44 (1976).
Review:  [For the entire collection see Zbl 327.00009.]
This is an entertaining survey paper on the factorization of factorials and binomial coefficients and related questions. Outlines of proofs of three new results are given. The first result states that to every \epsilon > 0 and \eta > 0 there is a k = k(\epsilon , \eta) so that the upper density of integers n for which the greatest prime factor of prod ki = 1(n+1) is less than n 1/2 - \epsilon is at most \eta. The third result says that for n > n0(\epsilon) the equation (*) n! = prod ti = 1(m+1) has no solutions for m < (2- \epsilon)n. It is conjectured that in the first result n 1/2 - \epsilon can be replaced by n1- \epsilon and that equation (*) has only trivial solutions except for 6! = 8.9.10. The constant mentioned in relation to Catalan's conjecture on consecutive powers and attributed to the reviewer should be read as an indication what kind of bound can be obtained by using the Gel'fond-Baker method. In the paper [Acta arithmetica 29, 197-209 (1976; Zbl 286.10013)] it is only proved that there exists an effectively computable constant.
Reviewer:  R.Tijdeman
Classif.:  * 11-02 Research monographs (number theory)
                   11A41 Elemementary prime number theory
                   11D57 Multiplicative and norm form diophantine equations
                   00A07 Problem books

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