Interactive Real Analysis - part of MathCS.org

Next | Previous | Glossary | Map | Discussion

Euclid Theorem:

There is no largest prime number.

Proof

Suppose 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: Is this new number M a prime number? We could check for divisibility: Hence, M is not divisible by a, b, c, ..., N. Since these are all possible prime numbers, M is not divisible by any prime number, and therefore M is not divisible by any number. That means that M is also a prime number. But clearly M > N, which is impossible, because N was supposed to be the largest possible prime number. Therefore, our assumption is wrong, and thus there is no largest prime number.

Next | Previous | Glossary | Map | Discussion