The Big Idea

So, after so many posts about numbers and factors, primeness and perfection, you might be wondering, what's the point? And you'd be very wise to ask such a question. There are two schools of thought in mathematics regarding the answer to this question. The majority of the world falls into the category of applied mathematicians, or people who believe that math isn't worth doing unless it has some practical application. The other portion believes in pure mathematics or the idea that math is interesting in and of itself, regardless of whether there is a practical application for it. Though I tend to be more applied than pure, I have elements of both in me.

However, in order to appease those of you who want to know the practical applications, I'll give you one very good one.

Prime numbers are the basis of the public key cryptography schemes used to encode information that is sent over the internet. Everytime you make a financial transaction of any kind, the information you send is encrypted before leaving your computer. It is then decrypted when it reaches its destination. The process used is based on the fact that multiplying numbers is VERY easy, while factoring them is VERY hard. A number is created by multiplying two VERY LARGE prime numbers together, and this new number is put out in the public (the public key) for use in encrypting messages. The creation of this number is very easy, because you just take any two prime numbers and multiply them together, and as we said before, multiplying is VERY easy. But to decrypt the message, you have to know the two prime numbers you started with. Only the person who created the public key knows these two numbers. Anyone else has to try to factor the very large public key in order to get to the two prime numbers, and as we just said, factoring is VERY hard. This makes it so that anyone intercepting your coded message can't decode it easily. In fact, with the size of the prime numbers being used to create these public keys, it is impractical even for a supercomputer to find the two factors of the public key, so your data is VERY safe.

Incidentally, the problem of factoring numbers has led computer scientists to develop faster and faster algorithms, as well as faster and faster circuitry to conquer the task more quickly.

Some questions arise out of this application of prime numbers. Is there an easy way to determine if a number is prime? (no) Is there a pattern to the distribution of the prime numbers? (no; although there is a pattern to the density of prime numbers). Are there formulas that can be used to generate prime numbers? (not any that work consistently).

One other note about pure vs. applied mathematics. Many areas of math were persued strictly for the sake of mathematics, and later on (sometimes as much as centuries later) those branches of math were seen to be useful to create new technologies. So, even though a practical application might not currently exist, one might be created in the future.