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Euclid's theorem

Euclid's theorem asserts that there is no largest 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.

We call this way of working a proof by contradiction.


Example 1

Assume that 5 would be the largest prime number. Multiplication of all prime numbers gives 2 × 3 × 5 = 30 and then you get 30 + 1 = 31. You can only divide this by itself and so it is a prime number. This simple calculation shows that there is no largest prime number.



The Greek mathematician Euclid described this theorem in 300 BC.

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