probability that two positive integers are relatively prime

The probability that two positive integers chosen randomly are relatively prime is


At first glance this “naked” result is beautiful, but no suitable definition is there: there isn’t a probability spaceMathworldPlanetmath defined. Indeed, the word “probability” here is an abuse of language. So, now, let’s write the mathematical statement.

For each n+, let Sn be the set {1,2,,n}×{1,2,,n} and define Σn to be the powerset of Sn. Define μ:Σn 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


Our statement is thus the following. For each n+, select random integers xn and yn with 1xn,ynn. Then the limit limnP((xn,yn)An) exists and


In other words, as n gets large, the fraction of |Sn| consisting of relatively prime pairs of positive integers tends to 6/π2.


  • 1 Challenging Mathematical Problems with Elementary Solutions, A.M. Yaglom and I.M. Yaglom, Vol. 1, Holden-Day, 1964. (See Problems 92 and 93)
Title probability that two positive integers are relatively prime
Canonical name ProbabilityThatTwoPositiveIntegersAreRelativelyPrime
Date of creation 2013-03-22 14:56:08
Last modified on 2013-03-22 14:56:08
Owner mps (409)
Last modified by mps (409)
Numerical id 21
Author mps (409)
Entry type Result
Classification msc 11A41
Classification msc 11A05
Classification msc 11A51