Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  251.10010
Autor:  Erdös, Paul
Title:  Über die Anzahl der Primfaktoren von \binom{n}{k}. (On the number of prime factors of \binom{n}{k}.) (In German)
Source:  Arch. der Math. 24, 53-56 (1973).
Review:  Let V(m) denote the number of different prime factors of m. H.Scheid [Arch. Math. 20, 581-582 (1969; Zbl 195.33001)] proved that for 2 < 2k \leq n

V \binom{n}{k} > {k log 2 \over log 2k}.

The author here proves the following theorem: For every \epsilon > 0 and k > k0(\epsilon), and for n \geq 2k,

V \binom{n}{k} > (1-\epsilon) {k log 4 \over log k}.

To show that the above result is in a sense accurate, he further proves V\binom{2k}{k} < (1+\epsilon){k log 4 \over log k}. An analogous proof is stated to hold for

V\binom{n}{k} < (1+\epsilon){n log 2 \over log n}.

Scheid considered it probable that, for fixed k, V\binom{n}{k} does not tend to infinity. The author in fact proves this statement to be true. If n > 2 · k!, he also proves that V\binom{n}{k} \geq k, and finally states the following conjecture. For almost all n < k1+\alpha

V\binom{n}{k} = (1+0(1))k log(1+\alpha).


Reviewer:  S.M.Kerawala
Classif.:  * 11A41 Elemementary prime number theory
                   05A10 Combinatorial functions

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