Download probability that two positive integers are relatively prime

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
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