# Filip Saidak's Proof

### By Chris Caldwell

Euclid may have been the first to give a proofthat there are infintely many primes. Below we give another proof by Filip Saidak [Saidak2005], similar to Goldbach's argument, but in a way even simpler.

**Theorem.**- There are infinitely many primes.
**Proof.**Let

*n*> 1 be a positive integer. Since*n*and*n*+1 are consecutive integers, they must be coprime, and hence the numberN

_{2}=*n*(*n*+ 1)must have at least two different prime factors. Similarly, since the integers

*n*(*n*+1) and*n*(*n*+1)+1 are consecutive, and therefore coprime, the numberN

_{3}=*n*(*n*+ 1)[*n*(*n*+ 1) + 1]must have at least 3 different prime factors. This can be continued indefinitely.∎

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