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### Proof by contradiction

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

##### Explanation

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.