Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 013.24603
Autor: Erdös, Paul
Title: On the integers which are the totient of a product of three primes. (In English)
Source: Q. J. Math., Oxf. Ser. 7, 16-19 (1936).
Review: The main result of this paper is that if f(n) is the number of representations of n as (p-1) (q-1) (r-1) where p,q,r are different primes, then limsupn ––> oo f(n) = oo. The proof is based on a slightly more precise form of the proposition (established by the author in a previous paper, Zbl 012.14905) that for almost all primes p \leq n, p-1 has between (1-\epsilon) log log n and (1+\epsilon) log log n different prime factors. He is then able to prove that, if p'q'r' denote primes with p'q'r' \leq n for which p'-1, q'-1, r'-1 have each more than (1-\epsilon) log log n different prime factors, then the number of different number of the form (p'-1) (q'-1) (r'-1) is of lower order of magnitude than the number of sets p'q'r'. This establishes the main result. Correction. The formula between (2) and (3) should read: N(Pk,n) < {Cn\over log2 n} {(C+ log log n)k+3\over (k-1)!}+o({n\over log2 n}).
Reviewer: Davenport
Classif.: * 11A25 Arithmetic functions, etc.
11A41 Elemementary prime number theory
Index Words: Algebra, number theory
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