MathCS.org - Real Analysis: Euclid Theorem:

Euclid Theorem:

There is no largest prime number.

Context

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

Is this new number 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
.....

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.

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Euclid Theorem:

Proof