Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  313.10045
Autor:  Erdös, Paul
Title:  Remarks on some problems in number theory. (In English)
Source:  Math. Balk. 4, 197-202 (1974).
Review:  The paper, presented at the fifth Balkan Congress, held in Beograd in 1974, consists of three parts in each of which a different subject is treated. Part I considers the problem of finding an estimate from above for S(x), the number of integers n \leq x for which there is a non-cyclic simple group of order n. Let V denote the sequence of integers v1 < v2 ... with the property that for every prime p/vi, vi has a divisor di \equiv 1 (mod p), di > 1. Further U denotes the sequence of integers (ui) for which this property holds at least for the largest prime dividing ui. If V(x) and U(x) denote the number of integers not exceeding x in the respective sequences, then S(x) \leq V(x) \leq U(x). It is then proved that

U(x) < x \exp ((- 1/2 +0(1))(log x log log x) ½),

which improves an earlier result of Dornhoff and Spitznagel for S(x). The main tool is de Bruijn's well-known result on the number of integers not exceeding x, all whose prime factors are not greater than y. It is conjectured that

V(x) = x \exp (-(1+0(1)) c5(log x) ½ log log x).

Part II discusses a problem due to Hadwiger: Let D(n) denote the set of integers with the property that if k in D(n) then the n dimensional unit cube can be decomposed into k homothetic n dimensional cubes. Let c(n) be the smallest integer such that k \geq c(n) implies k in D(n). It is proved that

c(n) \leq (2n-2)((n+1)n-2)-1

and a number of related number theoretical problems are discussed.
Part III is devoted to the functions \sigma and \phi. Several previous results of the author are mentioned and, as goes without saying for a paper of Erdös's, various unsolved problems are stated. Finally, this part of the paper contains some new results with hints of their proofs. We mention one:

\nu (\sigma (n)) = (1/2 +0(1))(log log n)2,

with \nu the function that counts the different prime factors of n.
Reviewer:  H.Jager
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11A25 Arithmetic functions, etc.
                   11-02 Research monographs (number theory)
                   20D05 Classification of simple and nonsolvable finite groups
                   11H99 Geometry of numbers
                   00A07 Problem books

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