Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  393.10046
Autor:  Erdös, Paul
Title:  Some unconventional problems in number theory. (In English)
Source:  Acta Math. Acad. Sci. Hung. 33, 71-80 (1979).
Review:  This fascinating miscellany covers a great many topics and the best this reviewer can do is attempt to give a taste of the paper. As stated by the author the paper mostly deals with arithmetic functions, primes, divisors, sieve processes and consecutive integers. Let f be an arithmetic function. The integer n is called a barrier for f if m+f(m) \leq n for every m < n. For n = prod pi\alphai put d0(n) = prod\alphai. Then d0(n) has infinitely many barriers, that is there are infinitaly many n such that m+d0(m) \leq n for every m < n. In fact the density of integers n which are barriers for d0 is positive. An outline of the proof of this Theorem is given including the following observation. Let \epsilon > 0, k be a sufficiently large integer and A be a multiple of p1,p2,...,pk. Then the density of integers t for which d0(tAk-i) > i, for some i with l \leq i \leq k, is less than ½ \epsilon. Among others the following problem is discussed. Let p(m) denote the least prime factor of m and put F(n) = max{m+p(m): l \leq m < n, m composite}. Is it true that F(n) \leq n for infinitely many n?