Question

Prove that there are infinitely many primes

Answer 1

Let us assume that there are finitely many primes and label them $p_1,\dots,p_n$. We will now construct the number $P$ to be one more than the product of all finitely many primes: $P=p_1p_2\dots p_n+1$.

The number $P$ has remainder $1$ when divided by any prime $p_i$, $i=1,\dots,n$, making it a prime number as long as $P\ne1$.

Since $2$ is a prime number, the list of $p_i$'s is non-empty. It follows that $P$ is greater that one and so has two distinct divisors. It is therefore a prime number.

It can also be seen from the definition of $P$ that it is strictly greater than any of the $p_i$'s.  This contradicts our assumption that there are finitely many prime numbers. Therefore, there are infinitely many prime numbers.