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Euclid's theorem proved that there are an infinite number of primes.

Imagine the biggest prime number ever discovered (call it P). Then take P factorial and add 1. Call this new number P! + 1 = Q. What's interesting is that no number between 1 and P will divide perfectly into Q, since all numbers between 1 and P divide perfectly into P factorial, and Q is P factorial + 1. Therefore any prime factors Q may have are either greater than P, proving there is at least one prime greater than P, or Q has no factors at all besides itself and 1, also proving there is at least one prime greater than P (namely Q itself). In fact it's not only at least one, but at least infinity, since any prime number greater than the original P can itself represent P.

Hopefully this makes sense.

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