On a conjecture of Balog
Author:
Adolf Hildebrand
Journal:
Proc. Amer. Math. Soc. 95 (1985), 517523
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 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 .
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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
(13,724e)
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P.
Erdős, Problems and results on number theoretic properties
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Congressus Numerantium, No. XVI,Utilitas Math. MR 0422146
(54 #10138)
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D.
R. HeathBrown, The divisor function at consecutive integers,
Mathematika 31 (1984), no. 1, 141–149. MR 762186
(86c:11071), http://dx.doi.org/10.1112/S0025579300010743
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 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), 5060. 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. 2544. MR 0422146 (54:10138)
 [4]
 D. R. HeathBrown, The divisor function at consecutive integers, Mathematika 31 (1984), 141149. MR 762186 (86c:11071)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198508101554
PII:
S 00029939(1985)08101554
Article copyright:
© Copyright 1985
American Mathematical Society
