Maximum Gcd among Pairs of Random Integers
نویسندگان
چکیده
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 N2−η and N2+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 N2 / log(N). The proof generalizes to any arithmetical semigroup where a suitable form of the Prime Number Theorem is valid.
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