Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 339.10006
Autor: Erdös, Paul; Gupta, H.; Khare, S.P.
Title: On the number of distinct prime divisors of \binom{n}{k}. (In English)
Source: Utilitas Math. 10, 51-60 (1976).
Review: For positive integers n,k; V(n,k) denotes the number of distinct prime divisors of \binom{n}{k}; mk is the least n for which V(n,k) = k; nk is the least n for which V(n,k) \geq k; and Nk is the least integer such that for every n \geq Nk, V(n,k) \geq k. It is likely (but not certain) that mk exists for each k. The authors prove: (i) for n > 4939, nk > k2; (ii) for n > 4949, nk > k2 ln k; (iii) for k > k0(c), nk > ck2 ln k; (iv) liminfk ––> oo (ln nk/ ln k) \leq e; (v) for every \epsilon > 0 and k > k0(\epsilon), Nk < (e+\epsilon)k. In Math. Scandinav. 39, 271-281 (1976; Zbl 344.10003) and Ernst S.Selmer has given nk for each k \leq 200. From his results, it appears that the only values for which nk \leq kk are k = 2,3, ... ,30,32,36,37. The tables in the paper under review, record some interesting facts about \binom{n}{k}, e.g. \binom{23}{k} is a product of distinct primes for each k \leq 22. Moreover mk is not always < mk+1 nor is nk necessarily \geq mk. Besides Selmer's paper, the reader might also refer to the reviewer's paper [Publ. Fac. Électrotechn. Univ. Belgrade, Sér. Mat. Phys. 498-541, 77-83 (1975; Zbl 315.10004)].
Classif.: * 11A41 Elemementary prime number theory
11B39 Special numbers, etc.
05A10 Combinatorial functions
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