Probability, Primes, and Pi

What is the probability that two randomly chosen integers have no prime factors in common?  In honor of Pi Day, I’d like to explain the surprising answer: 6/\pi^2.

The hero of this story is Leonhard Euler, who worked out this astonishing connection between prime numbers and \pi through a series of brilliant insights.  In the spirit of Euler, I will be rather cavalier about issues of convergence and rigor here, focusing on the key underlying ideas.

18th century mathematician Leonhard Euler

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