Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 473.10004
Autor: Erdös, Pál
Title: On prime factors of binomial coefficients. (In Hungarian)
Source: Mat. Lapok 28, 287-296 (1980).
Review: Denote P(m) the greatest, while p(m) the smallest prime divisor of the natural number m. Let \binom{n}{k} = UVW, where P(U) \leq k, P(V) > k, but P(V) \leq n-k, and W = prodn-k < p \leq np. This paper deals with M = max{U; V; W}, assuming n \geq 2k. The following results are proven. For k > 3, M = U holds only for finitely many pairs n,k. Let \delta > 0. If k \geq k0(\delta) and n < (14/3-\delta)k then M = W, but if n > (14/3+\delta)k then M = V. Furthermore if k \geq 10 or k = 6,8 and n > n0(k) then M = V. Finally for any integer r there exist infinitely many pairs n,k such that U(n,k) > W(n,k) > nr. The results are ineffective, because in the proofs an ineffective result of K. Mahler concerning the greatest prime divisor of binomial coefficients is used. In the paper one can also find some questions and problems.
Reviewer: A.Pethö
Classif.: * 11A05 Multiplicative structure of the integers
05A10 Combinatorial functions
11A41 Elemementary prime number theory
Keywords: smallest prime divisor; greatest prime divisor; ineffective results
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