## 1.1. Notation and Set Theory

### Euclid Theorem:

### 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:

*M = a * b * c * ... * N + 1*

*M*a prime number? We could check for divisibility:

*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*- .....

*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.