## 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 u1 = 1 < u2 < u3 < ··· be the sequence of integers of the form x2+y2. It is immediate, as shown by Bambah and Chowla, that ui+1-ui < cu^ 1/4 i. The conjecture ui+1-ui = o(u^ 1/4 i) is still improved. Turán observed to Erdös that ui+1-ui > c log ui/ log log ui for infinitely many i.
The author improves Turán's result to: ui+1-ui > c log ui/(log log ui)^ 1/2 . More generaly be proves that if p1 < p2 < ··· is a sequence of primes such that sumpi \leq x 1/p f(x) ––> oo as x ––> oo, and vi < v2 < ··· denote the integers wich either are not divisible by pi or are divisible by p2i, then for infinitely many i vi+1-vi > c e(f log vi) log vi/ log log vi. 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag