Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  541.10002
Autor:  Erdös, Pál
Title:  On prime factors of binomial coefficients. II. (In Hungarian)
Source:  Mat. Lapok 30, 307-316 (1982).
Review:  [For part I, cf. the author and R. L. Graham, Fibonacci Q. 14, 348- 352 (1976; Zbl 354.10010).]
This paper contains some results and unsolved problems concerning the prime factorization of the binomial coefficient \binom{n}{k}. Let V(n,k) be the contribution of prime powers pa to \binom{n}{k} with k < p \leq n-k. It is proved that if m(n) denotes the greatest number satisfying V(m(n),k) \leq V(n,k) then m(n) >> n1+1/k. Let f(k) resp. F(k) be the smallest resp. greatest number satisfying \omega(\binom{f(k)}{k}) \geq k resp. \omega(\binom{F(k)}{k}) < k. Is it true that F(k) > f(k)? Is it true that log f(k) = (1+o(1)) e log k? It is proved that F(k) \leq Ak, where Ak is the smallest common multiple of the integers up to k. Is it true that F(k) < \exp((1-c)k)?
Reviewer:  A.Balog
Classif.:  * 11A05 Multiplicative structure of the integers
                   05A10 Combinatorial functions
Keywords:  unsolved problems; prime factorization; binomial coefficient
Citations:  Zbl 354.10010

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