Why are there Infinite Primes?

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Did you know there are infinite prime numbers? Why are there infinite primes? Read on!

First to lay the groundwork, a number p is prime if:

a) p > 1
b) p has no positive divisors except 1 and p (itself)

Now let’s get into the infinite number of primes proof.

Let’s suppose q is the product of all primes up to pn plus 1
Expressed as a formula:

q = p1*p2*p3*…*pn + 1, where all of the pi‘s are prime

If we start plugging in numbers:

q = 2*3*5*…*pn + 1

Given this, we can say q is not divisible by any prime numbers ≤ pn
Why is that? It’s because of the “+ 1” we added to the end of q. Let’s say we didn’t have the + 1, then q would be divisible by every prime number up to pn, simply because it is a product of those numbers. The + 1 essentially forces a remainder, because 1 is smaller than the smallest prime, so if q is divided by any of the prime numbers used to make it, there will always be a remainder of 1.

Now let’s look at an example:

Let n = 3, then q = 2*3*5 + 1 = 31
if we divide 31 by 2, 3, or 5 we always get a remainder of 1

You can try this for any n, but you will always see that the product q is not divisible by any of the n primes used in the product.

This means that either q itself is a prime number,
or if it is not a prime number there must exist a prime number pk that q is divisible by, where pn < pk ≤ q

One the above has to be true, and in either case, it requires there to exist a prime number, either q or pk, that is greater than pn. Since n can be anything, there must be infinite primes, because we can always find a prime larger than the nth prime.

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