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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Spooky inference and the axiom of choice

A large crowd had gathered in Harvard Square, and I was curious what all the cheering and gasping was about.  HarvardSquareWorking my way through the crowd, I saw a street performer who (according to the handwritten red letters on his cardboard sign) went by the name “Zorn the Magnificent”.  He displayed a large orange, borrowed an extremely sharp knife from his assistant, and proceeded to chop the orange into five exotic-looking pieces while standing on one hand.  Working with almost unfathomably deft precision, he rearranged the pieces into two oranges, each the same size as the original one.  The oranges were given out for inspection and the crowd cheered wildly.  I clapped as well — even though I was familiar with the old Banach-Tarski paradox — since it was nevertheless an impressive display of skill and I had never seen it done one-handed before.  I heard a man with a long white beard whisper to the woman next to him “He hides it well, but I know that he’s secretly using the Axiom of Choice.” Continue reading