Question

Show that there are infinitely many positive primes.

Answer 1

Let us assume that there are a finite number of positive primes $p_1,p_2,,...,p_n$ such that $p_1<p_2<p_3<...<p_n$

Let $x=1+p_1{p}_2{p}_3...p_n$

We know that $p_1{p}_2{p}_3...p_n$ is divisible by each of $p_1,p_2,p_3<...<p_n$

But $x=1+p_1{p}_2{p}_3...p_n$ is not divisible by any one of $p_1,p_2,...,p_n$

$\therefore x$ is a prime number or we can say that it has prime divisors other then $p_1,p_2,...,p_n$

There exists a positive prime divisor other than $p_1,p_2,...,p_n$

This contradicts our assumption that there are a finite number of positive primes.

Hence, the number of positive primes is infinite.