## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  563.10002
Autor:  Erdös, Paul
Title:  Miscellaneous problems in number theory. (In English)
Source:  Numerical mathematics and computing, Proc. 11th Manitoba Conf., Winnipeg/Manit. 1981, Congr. Numerantium 34, 25-45 (1982).
Review:  [For the entire collection see Zbl 532.00008.]
Let n! = prodpipi\alphai(n) be the prime factor decomposition of n! into distinct prime powers. J.L.Selfridge and the author proved the interesting Theorem. Denote by h(n) the number of distinct exponents \alphai(n). There are absolute positive constants c1 and c2 for which

c1(n/ log n) ½ < h(n) < c2(n/ log n) ½.

The author conjectures that there exists a constant c > 0 such that h(n) = (c+o(1))(n/ log n) ½. Then he makes some conjectures about the prime factor decomposition of prodni = 1(x+i).
Next he proves the following Theorem. Let (1+\epsilon)n < a1 < a2 < ... < ak, (a1...ak)/n! = In where In has all its prime factors \leq n. Further let ak-a1 < n. Then a1 > 2n-c3nL where L = log log n/ log n. Finally some results on additive number theory are given.
Reviewer:  K.Ramachandra
Classif.:  * 11-02 Research monographs (number theory)
11A41 Elemementary prime number theory
11N37 Asymptotic results on arithmetic functions