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.

Fermat Primes are primes in the form of
Fermat (http://en.wikipedia.org/wiki/Fermat) conjectured that this is always prime for all n.

Then Euler proved that for n=5, it is divisible by 641. Now, all Fermat Numbers have been proven up to (i.e. n = 2167797 in the equation ) and all are in the form .

Now the question is, "is composite for all ?"

My idea is to find a correlation between the prime factor n and the exponent of the Fermat Prime? And if so, then I can prove using Sierpinski Numbers (http://en.wikipedia.org/wiki/Sierpinski_number) which are numbers in the form for all odd ks. So if you cannot get a prime factor from that form, then there is no prime factor for that Fermat Number, and if I can prove the opposite, then I can prove that there in no Fermat Prime for n>4!

Best of Luck on this Problem!
Zach

First Post

Hey all,

This is my first posting of my new MATH BLOG!!! I got the idea from a friend of mine, so my plan is to write about things I'm interested in/reading about/trying to solve. I hope you all enjoy!

Zach