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Proof by contradiction

With a "proof by contradiction" you can show that there is no greatest prime number.



If there is a finite number of primes, then you can determine the product P of all prime numbers.

Now you might ask: Is P + 1 a prime number?

The answer is "no", because we have already used all primes to calculate P. But you can also answers "yes", because you can divide P by any prime, and for P + 1 that is definitely not possible. So P + 1 itself must be a prime number. But that is completely contradictory with the starting point in which it was stated that there would exist a finite number of primes.

Our conclusion must be that there are infinitely many primes, and so there is no largest prime number.


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