Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a conjecture of Balog

Author: Adolf Hildebrand
Journal: Proc. Amer. Math. Soc. 95 (1985), 517-523
MSC: Primary 11A05; Secondary 11B05
MathSciNet review: 810155
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A conjecture of A. Balog is proved which gives a sufficient condition on a set $ A$ of positive integers such that $ A \cap (A + 1)$ is infinite. A consequence of this result is that, for every $ \varepsilon > 0$, there are infinitely many integers $ n$ such that both $ n$ and $ n + 1$ have a prime factor $ > {n^{1 - \varepsilon }}$.

References [Enhancements On Off] (What's this?)

  • [1] A. Balog, Problem in Tagungsbericht 41 (1982), Math. Forschungsinstitut Oberwolfach, p. 29.
  • [2] N. G. de Bruijn, On the number of positive integers ≤𝑥 and free of prime factors >𝑦, Nederl. Acad. Wetensch. Proc. Ser. A. 54 (1951), 50–60. MR 0046375
  • [3] P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Winnipeg, Man., 1976, pp. 25–44. Congressus Numerantium, No. XVI,Utilitas Math. MR 0422146
  • [4] D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), no. 1, 141–149. MR 762186,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11A05, 11B05

Retrieve articles in all journals with MSC: 11A05, 11B05

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society