probability that two positive integers are relatively prime
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 , let be the set and define to be the powerset of . Define by . This makes into a probability space.
We wish to consider the event of some also being in the set . The probability of this event is
Our statement is thus the following. For each , select random integers and with . Then the limit exists and
In other words, as gets large, the fraction of consisting of relatively prime pairs of positive integers tends to .
- 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|
|Date of creation||2013-03-22 14:56:08|
|Last modified on||2013-03-22 14:56:08|
|Last modified by||mps (409)|