Here is the slickest proof that I have found, it is even more simple than Euclid's proof.
Consider two integers n and the next one n+1 where n>1. Form the product n×(n+1). Now it is true that two consecutive integers have no common factors so the integer nx(n+1) must have at least two different prime factors.Now form the product n x (n+1) x (n+2). By the same argument this integer must have at least three different prime factors. This process can be continued indefinitely so there must be an infinite number of primes.
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