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### Euclid's theorem

**Euclid's theorem** asserts that there is no largest 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.

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.

## HistoryThe Greek mathematician Euclid described this theorem in 300 BC. |