Quote:
Originally Posted by storm5510
I will admit it. When it comes to mathematical theory and advanced subjects, I am naive as one can be. I am not too proud to admit it. At the same time, I am fascinated by it. I am afraid this fascination came too late. I had no interest in it when I was younger, beyond what I needed in everyday life. Then the computer age came into being and I knew I had messed up, badly.
I like to think of myself as being a decent programmer. I took to it right off in college. Languages like Pascal, Basic, and COBOL were all easy. C+ and Assembly were a lot harder but I managed. The problem then, and still now, is my lack of mathematical understanding. This inability puts a cap on what I can do in programming.
I would like to get past my naivety to some small degree if possible. In the past when I have asked questions here, I have been given links to pages on Wikipedia, mostly. That is fine. There is no need to write something out that already exists. I found a lot of those pages using mathematical notation to varying degrees. Some of it I can understand; those being the things I saw when studying industrial and digital electronics in college. The rest, not so well.
So, here is the lay of it. In 2005, I wrote an application that could find prime numbers, in general. No specific types. It was the GIMPS project which peaked, and still holds, my interest. I knew that prime numbers were only divisible by themselves, and one. So, that is what I based my application on. Wikipedia calls what I used the "naive" way. It is the longest way; taking a number and dividing it by every odd value above two to the value of the number, minus two, and skipping units of five.
I would like to learn a better way to do this, and I am asking for assistance. If someone would lend a hand, that would be wonderful. If not, then that will be alright too. I understand.

Start by reading:
D.E. Knuth, The Art of Computer Programming, Vol 2.
This will teach you about multiprecise arithmetic and a lot of other things.
Aho, Hopcroft & Ullman also wrote an excellent book on Algorithms.
add a decent book on number theory.
Hardy & Wright, Introduction to the Theory of Numbers is a good
text and quite broad; it covers a lot of topics.
Try also:
D. Shanks, Solved & Unsolved Problems in Number Theory.
Then add a good general book about modern algebra. There are
a lot of good ones. Stay away from S. Lang's effort.
I would recomment Hungerford's book, but it is likely too difficult.
Birkhoff & MacLane is excellent.
Then you can try reading Crandall & Pomerance's book.
Read. Read. Read. Do the exercizes. There is no shortcut.