There is no largest prime number.
ProofSuppose there was a largest prime number; call it N. Then there are only finitely many prime numbers, because each has to be between 1 and N. Let's call those prime numbers a, b, c, ..., N. Then consider this number:
- M = a * b * c * ... * N + 1
- M is not divisible by a, because M / a = b * c * ... * N + 1 / a
- M is not divisible by b, because M / b = a * c * ... * N + 1 / b
- M is not divisible by c, because M / c = a * b * ... * N + 1 / c