3
$\begingroup$

Assume that in the group of integers $\mathbb{Z}$, I randomly choose two integers $a$ and $b$ and I would like to ask whether they generate the group $\mathbb{Z}$, and then, what is the probability for this event? However, to explain clearly "probability meaning", we need to know about probability distribution. Could any one give me an exact definition for this then? Or here, we just reduce our consideration on each quotient group $\mathbb{Z}/n\mathbb{Z}$ for each $n$?

Thanks in advance.

  • 0
    Sorry, $6/\pi^2$.2012-11-30

1 Answers 1

11

There is no uniform distribution on the integers.

But if you take two random integers uniformly from the interval $[-n,n]$, then the limit of the probability that they are coprime is $\dfrac{1}{\zeta(2)}=\dfrac{6}{\pi^2} \approx 0.6079$.

  • 1
    Thank Henry, that's what I am thinking of.2012-11-30