Fermat’s numbers and infinity of primes

There are two extremely beautiful, fundamental theorems in mathematics — (1) the number of primes is infinite (2) the square root of 2 is irrational.

The oldest proof of the first theorem is due to Euclid. There is another proof of the first theorem due to Christian Goldbach (from a letter to Leonhard Euler, 1730). It is as follows: (Reference: Proofs from THE BOOK by Martin Aigner and Gunter M. Ziegler, fourth edition)

Let us look at Fermat Numbers $F_{n}=2^{2^{n}}+1$ for $n=0, 1, 2, \ldots$ . We will show that any two Fermat numbers are relatively prime; hence, there must be infinitely many primes. To this end, we verify the recursion

$\Pi_{k=0}^{n-1}F_{k}=F_{n}-2$ for $n \geq 1$

from which our assertion follows immediately. Indeed, if m is a divisor of $F_{k}$ and $F_{n}$ ( $k < n$ ), then m divides 2, and, hence, $m =1 \hspace{0.1in} or \hspace{0.1in} 2$ . But, m=2 is impossible since all Fermat numbers are odd.