Gauss, Number Theory Explained

Hey Everybody,

I recently bought the book, God Invented the Integers, by Stephen Hawkings. So, I decided that I will explain every paper to you(if I understand it, which I understand very little) So, I will start with Gauss.

Gauss is a very smart man. He created a new branch of mathematics called "number theory" He created a notation in his paper excerpted in the book. For example consider the equation:. This notation is very powerful. What this is saying, is that b is the remainder of a when divided by c. It is equivalent in finding an integer k that satisfies the equation . A number example may clear up confusion. For example, say we had the equation . Now, we solve the remainder and we get

.

This notation imbues a lot of different meanings into a simple format. For example, there is a Theorem, called Wilson's Theorem, that says p is a prime if and only if p evenly divides (where n! is the factorial function) In Gauss's notation this would be written as
. This notation is very powerful. What this is saying, is that b is the remainder of b when divided by c. It is equivalent in finding an integer k that satisfies the equation . A number example may clear up confusion. For example, say we had the equation . Now, we solve the remainder and we get , so the remainder of 10/4 is 2.

This notation imbues a lot of different meanings into a simple format. For example, there is a Theorem, called Wilson's Theorem, that says p is a prime if and only if p evenly divides (where n! is the factorial function) In Gauss's notation this would be written as

. (Where -1 means p-1) (I may offer up a proof of this later)

There are laws of congruences, such as




This is used in many fields of mathematics, including cryptography, and prime theories and others.

Next Entry:
Now that we got the basics out of the way, I will start on Gauss' paper, proof of the Fermat Little Theorem and laws of roots.