# Prime numbers go on forever

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!

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