Problem 1: How many prime numbers exist?

A prime number is any whole number that is not divisible by any number except itself and 1.

Examples of prime numbers are:  2,3,5,7,11,13, etc.

Any whole number that is a divisible by a smaller number greater than 1 is called a composite number. 

Thus, it follows that all whole numbers greater than 1 are either prime numbers or composite numbers.

Problem 1:  Prove that there are an infinite number of prime numbers.

Here are some hints (if needed):

(1)  If there is a finite number of prime numbers, then there is a largest prime number.

(2)  If there is a finite number of prime numbers, then the product of all prime numbers is itself a finite whole number.






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Solution 1: There are an infinite number of prime numbers

Here is the solution to Problem 1:  Prove that there are an infinite number of prime numbers. Note:  The argument makes use of proof by contradiction .  If you are not familiar with this type of argument, a tutorial can be found here and you can watch a video here . Claim:  There are an infinite number of prime numbers. (1)  Assume that there are only a finite number of prime numbers. (2)  Let $x$ be the largest such prime.   (3)  It follows that $x!$ is divisible by all primes.  Note:  $x!$ is the factorial for $x$.  If you are not familiar, you can learn about factorials here or see a video here . (4)  But then, no prime divides $x! + 1$ since all such divisions will have a remainder of 1. (5)  Thus, we have a contradiction and can reject our assumption in step 1. Either $x!+1 > x$ is a prime itself (which contradicts our assumption in step 1 that $x$ is the largest prime number) or $x!+1$ is composite and divisible by a prime number greater than $x$ (which also contradicts thi
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