Is there a finite number of primes? Or is the number of prime numbers infinite? 
 

Solution

The number of primes is infinite. Surprisingly, this is a very simple matter to prove, and the proof is accomplished by contradiction.

Assume that there is a finite number of prime numbers. Call them P₁, P₂,...P_n, where n is the number of primes.

Now multiply all these primes together, and add 1. This number is larger than all the primes, so it cannot be a prime. But it also cannot be written as a product of prime numbers, as it leaves a remainder of one when divided by each element of the primes. This is a contradiction, from which we conclude that there must be an infinite number of primes. 

Isn't that slick?

1.

Explain what a prime number is.

2.

If you are trying to find out if the number n is prime, you can try dividing it by all the integers from 1 to x. How big is x in terms of n?

3.

Find the smallest prime number bigger than 100.

4.

Find the largest prime number less than 100.