Alternative proof of infinitude of prime.

Question

I was chatting with my friends and one of them asked if the difference between 2 consecutive primes is increasing with increasing in natural numbers,, then will we have a point where all primes get finished?? i.e. will there exist another prime number so far from where we are??

I had no answers of his question at that moment.

But then I thought about this question and tried to prove that primes will never get finished or primes are infinte

My work:

Suppose there are finite number of primes and let $q$ is a number which is the product of all primes.

But $q+1$ is always co-prime with $q$ .

Hence no factors of $q$ will divide $q+1$.

So, we will get another prime number and the process will continue till infinity.

So we have proved that primes numbers are infinite.

I want to just verify that whether my prove is right or wrong.How can we prove it alternatively?

Answer 1

You have got two questions here:

1. Is my proof correct??

2. How can we prove it alternatively?

I have got answer for both.

$1$. Yes your proof is correct, though it shows a little deviation from the ancient Euclid's proof. For full proof go there.

$2$. Alternative Proof: I think this is the easiest one after Euclid's great proof.

Suppose there is an integer $Q_n=n!+1$.

Suppose that the largest prime which divides $Q_n$ is $p$.

Assume that $p\le n$. Then $p|n!$ annd we also had assumed that $p$ is the largest prime which divide $Q_n$ so $p|Q_n\implies p|n!+1$ This gives us $p|1$. Which is not possible. So, $p\ge n$.

Now, suppose that there is a largest prime $q$. But, by the previous proof we know that there is a prime larger than $q$ which divides $q!+1$, which means $q$ is not the largest prime $\implies$ A contradiction. Hence, Number of primes are infinite.

You can get some more proves here.