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 :

p1 , p2 , ..... pn

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

q = p1 * p2 * ..... * pn + 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