# Infinitely Many Primes?????

### By: Fabian Meraz

## 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:

## Sources:

http://spikedmath.com/292.html

__Discrete Math __Arthur T. Benjamin