A prime number is a natural number with exactly two distinct divisors:

1 and itself. Let us assume that there are nitely many primes and label them

p1; : : : ; pn. We will now construct the number P to be one more than the product

of all nitely many primes:

P = p1p2⋯pn +1:

The number P has remainder 1 when divided by any prime pi , i = 1; : : : ; n,

making it a prime number as long as P ≠ 1.

Since 2 is a prime number, the list of pi's is non-empty. It follows that P

is greater than one and so has two distinct divisors. It is therefore a prime

number.It can also be seen from the denition of P that it is strictly greater than

any of the pi's. This contradicts our assumption that there are nitely many

prime numbers. Therefore, there are innitely many prime numbers.

Alternatively, one can leave out the assumption and let p1; : : : ; n be any ar-

bitrary nite list of prime numbers. Then the conclusion would state that for

any nite list of prime numbers, it is possible to construct a larger prime than

any on the list. This method uses induction.