Prime numbers go on forever

The following proof goes back to the ancient mathematician Euclid (more precisely: Euclides of Alexandria).

We to take tentatively at, there would only be a finite number of prime numbers. If this were true, then there should be one largest of all prime numbers, and we denote this with n. The list of all prime numbers would then be Da & # 223, however this assumption cannot be correct, becomes evident when the number (i.e. the product all Prime numbers plus 1) is considered:

This number would be much larger than, so it could not be a prime number. Consequently it should have a divisor (different from 1 and itself). This factor could be broken down into a product of prime numbers, and all of these prime factors would have to divide the number (2). (If a number is divided by e.g. 10, then also by the prime factors 2 and 5). So there should be at least one prime that divides (2).

On the other hand (2) cannot be completely divided by any prime number in our list 2, 3, 5, ..., since the remainder always remains 1 !!! So there would be a prime that is not on our list! The contradicts but assuming that & # 223 we in (1) all Have listed prime numbers!

The assumption that there are only finitely many prime numbers leads to a (logical) contradiction, so it cannot be true! (Because the general rule is: a statement from which a contradiction can be constructed must be wrong).

This proves:

This way of proving a fact will ''indirect evidence'' called: If the assumption that the opposite of a statement is true can be used to construct a contradiction, then the statement must be true!