On a conjecture of Balog
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- by Adolf Hildebrand PDF
- Proc. Amer. Math. Soc. 95 (1985), 517-523 Request permission
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
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A. Balog, Problem in Tagungsbericht 41 (1982), Math. Forschungsinstitut Oberwolfach, p. 29.
- N. G. de Bruijn, On the number of positive integers $\leq x$ and free of prime factors $>y$, Nederl. Acad. Wetensch. Proc. Ser. A. 54 (1951), 50–60. MR 0046375
- 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) Congressus Numerantium, No. XVI,Utilitas Math., Winnipeg, Man., 1976, pp. 25–44. MR 0422146
- D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), no. 1, 141–149. MR 762186, DOI 10.1112/S0025579300010743
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 517-523
- MSC: Primary 11A05; Secondary 11B05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0810155-4
- MathSciNet review: 810155