Infinitely Many Primes?????

By: Fabian Meraz

I know that already

This questioned stumped many mathematicians for a long period of time. I am sure you already know that there are infinitely many primes. We don’t run out of primes. It was not until a mathematician, named Euclid, proved this that the question was finally answered.

Proof

We will prove this using proof by contradiction. Assume to the contrary that there are only finitely many prime numbers. Let p1, p2 ..., pn be the list of all the primes. Consider the number N = p1p2... pn + 1. The number N is either prime or composite. Note that N is congruent to 1 (mod pi) for each i = 1, 2, ..., n. By the fundamental theorem of arithmetic, N must factor into a product of primes. But no pi divides N, thus, N cannot be composite. We conclude that N is a prime number, not among the primes listed above, contradicting our assumption that all primes are in the list p1, p2 ..., pn.

How many primes are there up to the number n?

According to the prime number theorem, the fraction of numbers below n is approximately 1/[ln(n)-1]. So, if you wanted to approximate how many primes are less than n you would just multiply n by 1/ [ln(n)-1].

Test Question:

Approximately how many primes are less than 100?

Bonus Vid:

If you would like to see a more in-depth explanation of the proof, you should watch this video. While you are there, you should see their other videos; they are fun and interesting.
Infinite Primes - Numberphile