Paragraph

  1. Suppose there are only finitely many primes. [this is a premise. Note the use of “suppose.”]

  2. There must be a largest prime, call it \(p\text{.}\) [follows from line 1, by the definition of “finitely many.”]

  3. Let \(N = p! + 1\text{.}\) [basically just notation, although this is the inspired part of the proof; looking at \(p! + 1\) is the key insight.]

  4. \(N\) is larger than \(p\text{.}\) [by the definition of \(p!\)]

  5. \(N\) is not divisible by any number less than or equal to \(p\text{.}\) [by definition, \(p!\) is divisible by each number less than or equal to \(p\text{,}\) so \(p! + 1\) is not.]

  6. The prime factorization of \(N\) contains prime numbers greater than \(p\text{.}\) [since \(N\) is divisible by each prime number in the prime factorization of \(N\text{,}\) and by line 5.]

  7. Therefore \(p\) is not the largest prime. [by line 6, \(N\) is divisible by a prime larger than \(p\text{.}\)]

  8. This is a contradiction. [from line 2 and line 7: the largest prime is \(p\) and there is a prime larger than \(p\text{.}\)]

  9. Therefore there are infinitely many primes. [from line 1 and line 8: our only premise lead to a contradiction, so the premise is false.]

in-context