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Maximum GCD Among Pairs of Random Integers

R. W. R. Darling / E. E. Pyle
Published Online: 2011-04-11 | DOI: https://doi.org/10.1515/integ.2011.006


Fix α > 0, and sample N integers uniformly at random from {1, 2, . . . , ⌊eαN ⌋}. Given η > 0, the probability that the maximum of the pairwise GCDs lies between N 2–η and N 2+η converges to 1 as N → ∞. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order N 2/log(N). The proof generalizes to any arithmetical semigroup where a suitable form of the prime number theorem is valid.

Keywords.: Maximum pairwise greatest common divisor

About the article

Received: 2009-11-13

Revised: 2010-09-03

Accepted: 2010-11-23

Published Online: 2011-04-11

Published in Print: 2011-04-01

Citation Information: Integers, Volume 11, Issue 2, Pages 93–105, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integ.2011.006.

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