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probability that two positive integers are relatively prime∗ mps† 2013-03-21 18:31:31 The probability that two positive integers chosen randomly are relatively prime is 6 = 0.60792710185 . . . . π2 At first glance this “naked” result is beautiful, but no suitable definition is there: there isn’t a probability space defined. Indeed, the word “probability” here is an abuse of language. So, now, let’s write the mathematical statement. For each n ∈ Z+ , let Sn be the set {1, 2, . . . , n} × {1, 2, . . . , n} and define Σn to be the powerset of Sn . Define µ : Σn → R by µ(E) = |E|/|Sn |. This makes (Sn , Σn , µ) into a probability space. We wish to consider the event of some (x, y) ∈ Sn also being in the set An = {(a, b) ∈ Sn : gcd(a, b) = 1}. The probability of this event is Z |An | χAn dµ = P ((x, y) ∈ An ) = . |Sn | Sn Our statement is thus the following. For each n ∈ Z+ , select random integers xn and yn with 1 ≤ xn , yn ≤ n. Then the limit limn→∞ P ((xn , yn ) ∈ An ) exists and 6 lim P ((xn , yn ) ∈ An ) = 2 . n→∞ π In other words, as n gets large, the fraction of |Sn | consisting of relatively prime pairs of positive integers tends to 6/π 2 . References [1] Challenging Mathematical Problems with Elementary Solutions, A.M. Yaglom and I.M. Yaglom, Vol. 1, Holden-Day, 1964. (See Problems 92 and 93) ∗ hProbabilityThatTwoPositiveIntegersAreRelativelyPrimei created: h2013-03-21i by: hmpsi version: h36625i Privacy setting: h1i hResulti h11A41i h11A05i h11A51i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1