# A proof via Lagrange’s theorem

May 22, 2012 Leave a comment

Let’s assume for the sake of contradiction that there are finitely many primes. Let denote the biggest prime. Since , it cannot be prime. It must be divisible by some other prime . Another way to say this is that

This means that the order of in the multiplicative group must be a divisor of . BUT IS PRIME! So must have order . But then Lagrange’s theorem says that must be a divisor of the size of the group (which is ). In particular, , contradicting the “biggest-ness” of .

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