##
**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! = **prod**_{pi}p_{i}^{\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 \alpha_{i}(n). There are absolute positive constants c_{1} and c_{2} for which c_{1}(n/ log n)^{ ½} < h(n) < c_{2}(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 **prod**^{n}_{i = 1}(x+i).

Next he proves the following Theorem. Let (1+\epsilon)n < a_{1} < a_{2} < ... < a_{k}, (a_{1}...a_{k})/n! = I_{n} where I_{n} has all its prime factors \leq n. Further let a_{k}-a_{1} < n. Then a_{1} > 2^{n-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

11B13 Additive bases

11P99 Additive number theory

00A07 Problem books

**Keywords: ** disjoint sets of positive integers; distinct sum; unconventional problems; consecutive integers; factorial; prime factor decomposition; prime factors

**Citations: ** Zbl 532.00008

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag