On a conjecture of Balog

Author:
Adolf Hildebrand

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

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Abstract: A conjecture of A. Balog is proved which gives a sufficient condition on a set of positive integers such that is infinite. A consequence of this result is that, for every , there are infinitely many integers such that both and have a prime factor .

**[1]**A. Balog, Problem in Tagungsbericht**41**(1982), Math. Forschungsinstitut Oberwolfach, p. 29.**[2]**N. G. de Bruijn,*On the number of positive integers**and free of prime factors*, Nederl. Akad. Wetensch. Proc. Ser. A**54**(1951), 50-60. MR**0046375 (13:724e)****[3]**P. Erdös,*Problems and results on number theoretic properties of consecutive integers and related questions*, Proc. Fifth Manitoba Conf. on Num. Math. (Univ. Manitoba, Winnipeg, Man., 1975). Congressus Numeratium, no. XVI, Utilitas Math., Winnipeg, Man., 1976, pp. 25-44. MR**0422146 (54:10138)****[4]**D. R. Heath-Brown,*The divisor function at consecutive integers*, Mathematika**31**(1984), 141-149. MR**762186 (86c:11071)**

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0810155-4

Article copyright:
© Copyright 1985
American Mathematical Society