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Prove that there are infinitely many primes.

Let q=2\cdot 3\cdot 5\cdot \cdot \cdot p+1 where p is a prime. Then q is not divisible by any prime less than or equal to p. Now either q is prime or it is divisible by primes greater than p and less than q. In either case there is a prime greater than p, which proves the theorem.


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