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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 ≤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**, https://doi.org/10.1112/S0025579300010743

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

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

Additional Information

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

Article copyright:
© Copyright 1985
American Mathematical Society