Prime numbers are one of the more fascinating things in mathematics. The fact that they are numbers that are only evenly divisible by one and themself seems like quite an arbitrary definition. Even more fascinating in that they have no set pattern for when they occur. Many attempts have been made to find a pattern but none have been successful. One of the more impressive attempts was made by the incredible mathematician Riemann (who was a student of Gauss, whose work is used extensively in electromagnetism, for example in calculating magnetic flux). Riemann had a hypothesis that estimated the occurance of prime numbers, but it has never been completed and their is still a reward for proof of his hypothesis. The book 'Prime Obsession' by John Derbyshire is an excellent extenisve review of primes and is one of the best science books I have read in years.
Many times the study of higher mathematics has no practical application in the real world. Many mathematicians even brag about this fact. This was initially true of the study of prime numbers, for thousands of years, until within the last few decades they have taken on extreme importance in computer security. As is mentioned in the introduction to this section, multiplying two primes will give you a product, which you can give out as a 'public key'. It will be nearly impossible to factor this product into the two primes that created it. These two primes are the 'private key'. The importance of security in computers today cannot be understated, and its amazing to think it would not be possible were it not for this seemingly pointless pursuit. They will never have to travel over the internet and will thus never be exposed to being intercepted. The product encrypts the data, and the two primes decrypt it. There are many other fascinating examples in Mathematics of a study of a particular subject that was purely academic and served no purpose at the time, but the knowledge was later used in something practical. This is a justification for funding and pursuing all mathematical pursuits, even ones with no obvious purpose.