# Mathematics Weblog

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## Amazing Formula

Monday 11 October 2004 at 9:47 pm | In Articles | 3 CommentsThere are many interesting formulae in mathematics;

must be one of the most amazing of all.

The first reaction is where did that come from? You can find 14 different proofs of this in a paper on Robin Chapman’s Home Page [*look for Evaluating zeta(2)]*

Given this result can you prove another amazing result?

*If you pick two positive integers at random, the probability of them having no common divisor is *

gets everywhere! See Wikipedia for more such as

- or or

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I don’t know if your probability makes sense…

How do you pick two positive integers at random?

Is it possible to pick at random from an infinite set?

Comment by Ronald — Sunday 14 November 2004 3:48 pm #

Have a look at http://www.physics.harvard.edu/probweek/sol44.pdf where they make the statement more precise:

To be precise about what we mean by probabilities in this problem, we really shouldword the question as: Let N be a very large integer. Pick two random integers less

than or equal to N. What is the probability that these numbers are relatively prime,

in the limit where N goes to infinity?

Comment by Steve — Sunday 14 November 2004 4:10 pm #

That’s clever… thank you.

Comment by Ronald — Thursday 23 December 2004 12:18 pm #