Theorem:
Euclid's 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:
- M = a * b * c * ... * N + 1
Is this new number M a prime number? We could check for divisibilty:
- 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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