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**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 c_{1} such that (p_{n} denoting the n-th prime) for an infinity of n p_{n+1}-p_{n} > c_{1} {log p_{n} log log p_{n} \over (log log log p_{n})^{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 p_{1},...,p_{m} and with

z < p_{1}...p_{m}, \,\, l > {c_{2}p_{m} log p_{m} \over (log log p_{m})^{2}}. The proof depends on Brun's method and on an ingenious division of primes into classes. The main result follows on taking p_{m} to be the prime next below ^{1}/_{2} log p_{n}.

**Reviewer: ** Davenport (Cambridge)

**Classif.: ** * 11N05 Distribution of primes

**Index Words: ** Number theory

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