A simple proof of the infinitude of primes

The following is a simplification of the proof of the infinitude of primes using continued fractions given by Barnes [1]. Assume that there are only finitely many prime numbers, namely 2, p1 = 3, . . . , pn. Let q = p1 · · · pn be the product of all odd primes; then q + 1 is not divisible by any odd prime, hence must be a power of 2. Since q +1 ≡ 2 mod 4, we must have q +1 = 2 and therefore q = 1: contradiction. Since no odd prime p ≡ 3 mod 4 can divide q + 1, the proof actually shows that there are infinitely many primes p ≡ 1 mod 4.