#### Yes, there are infinitely many primes

The following proof, which is due to Euclid of Alexandria, is based on contradiction, that is, we claim that there are only a finite number, *n*, of them. We order them and write them down :

p_{1} , p_{2} , ..... p_{n}

We now form the product of all of these primes and add 1 :

q = p_{1} * p_{2} * ..... * p_{n} + 1

According to our claim the number *q* thus calculated is not prime, or in other words, we claim that it is divisible by two or more of the *n*
known primes. But that is clearly not the case, because *q* divided by any of the known *n* primes gives always a remainder of 1 according to the above equation.

*Fallacy :*If you do an example calculation *q* based on the
"known" primes 2, 3, 5, 7, 11, and 13 you get *q=30031* which is divisible by 59 and 509. Does that ruin our proof and why not ?

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Last revised: 10/23/13