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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 044.03604

**Autor: ** Erdös, Pál

**Title: ** Some problems and results in elementary number theory. (In English)

**Source: ** Publ. Math., Debrecen 2, 103-109 (1951).

**Review: ** Let u_{1} = 1 < u_{2} < u_{3} < ··· be the sequence of integers of the form x^{2}+y^{2}. It is immediate, as shown by Bambah and Chowla, that u_{i+1}-u_{i} < cu^ ^{1}/_{4} _{i}. The conjecture u_{i+1}-u_{i} = o(u^ ^{1}/_{4} _{i}) is still improved. Turán observed to Erdös that u_{i+1}-u_{i} > c log u_{i}/ log log u_{i} for infinitely many i.

The author improves Turán's result to: u_{i+1}-u_{i} > c log u_{i}/(log log u_{i})^ ^{1}/_{2} . More generaly be proves that if p_{1} < p_{2} < ··· is a sequence of primes such that **sum**_{pi \leq x} ^{1}/_{p} f(x) ––> oo as x ––> oo, and v_{i} < v_{2} < ··· denote the integers wich either are not divisible by p_{i} or are divisible by p^{2}_{i}, then for infinitely many i v_{i+1}-v_{i} > c e^{(f log vi}) log v_{i}/ log log v_{i}. In the last part of the paper the author gives some results concerning consecutive squarefree numbers. The relations (5), (10), (11) and (28) contain some misprints.

**Reviewer: ** Sigmund Selberg

**Classif.: ** * 11N25 Distribution of integers with specified multiplicative constraints

00A07 Problem books

**Index Words: ** number theory

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