Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  012.01102
Autor:  Erdös, Pál
Title:  On the difference of consecutive primes. (In English)
Source:  Q. J. Math., Oxf. Ser. 6, 124-128 (1935).
Review:  The author proves that there exists an absolute constant c1 such that (pn denoting the n-th prime) for an infinity of n

pn+1-pn > c1 {log pn log log pn \over (log log log pn)2},

this being an appreciable improvement on previous results (see E.Westzynthuis Zbl 003.24601 and G.Ricci Zbl 010.24801). It is first proved that for any m, one can find consecutive integers z,z+1,...,z+l each of which is divisible by at least one of p1,...,pm and with

z < p1...pm, \,\, l > {c2pm log pm \over (log log pm)2}.

The proof depends on Brun's method and on an ingenious division of primes into classes. The main result follows on taking pm to be the prime next below 1/2 log pn.
Reviewer:  Davenport (Cambridge)
Classif.:  * 11N05 Distribution of primes
Index Words:  Number theory

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