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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
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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, 10.1112/S0025579300010743

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DOI: https://doi.org/10.1090/S0002-9939-1985-0810155-4
Article copyright: © Copyright 1985 American Mathematical Society