A Simple Proof of Wilson’s Theorem
This elegant eighteenth century theorem gives a necessary and sufficient condition for proving primality
I remember when I first learned of Wilson’s theorem while reading Calvin Clawson’s marvelous book Mathematical Mysteries: The Beauty and Magic of Numbers. I was awestruck by the fact that such a simple formula could do something so important — stating the necessary and sufficient conditions for primality. I needed to learn precisely why it works.
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Origins of the Theorem
John Wilson was an English mathematician and judge who is best known for the formula we are about to prove. While the theorem is named for him, it was published by his teacher, Edward Waring in 1770 and subsequently proved by Joseph-Louis Lagrange in 1771. It was apparently known by the Persian mathematician Ibn al-Haytham around the end of the first millennium.
Simply put, Wilson’s theorem states that a number, p, is prime if and only if p divides (p -1)! +1:
