Proof of LL theorem

[R.D. Silverman]

Originally Posted by John Renze

It is possible to prove the LL test directly by simpler means,
John

You are totally misusing the word "simpler". The alternative proofs are
NOT simpler. They are much more complex and much longer. They just
avoid the use of elementary abstract algebra. However, the group-
theoretic proof is much simpler, cleaner, and conveys much more UNDERSTANDING than the long, complicated modular arguments. However,
it does require knowing a little basic group theory.

[R.D. Silverman]

Originally Posted by John Renze

"Nothing much above first year calculus level" has a plain meaning that I understood immediately.

Please explain what you think "above first year calculus" really means.

[Orgasmic Troll]

Originally Posted by R.D. Silverman

You are totally misusing the word "simpler". The alternative proofs are
NOT simpler. They are much more complex and much longer. They just
avoid the use of elementary abstract algebra. However, the group-
theoretic proof is much simpler, cleaner, and conveys much more UNDERSTANDING than the long, complicated modular arguments. However,
it does require knowing a little basic group theory.

I'm halfway through my abstract algebra series (Up through chapter 6 in Artin's Algebra, if you're familiar with it) I know Lagrange's theorem, however, I don't think we've gone over multiplicative sub-groups of a finite field just yet. Would you mind going over the proof with some pointers of where I should do my homework to understand the bits I don't know?

[mfgoode]

Alpertron: if I am permitted to comment on mersennewiki ,
I checked out the LL test fully and found the print too small. After a while of reading it was hurtful to the eyes. I have seen some articles where a built in magnifying power by clicking is accomplished. So if you could incorporate this feature or increase the size even to this print one can sit longer on mersennewiki.
I think this feature of magnifying is included in Amazon,com for reviewing books in small print.
Mally

[John Renze]

Originally Posted by R.D. Silverman

You are totally misusing the word "simpler". The alternative proofs are
NOT simpler. They are much more complex and much longer. They just
avoid the use of elementary abstract algebra. However, the group-
theoretic proof is much simpler, cleaner, and conveys much more UNDERSTANDING than the long, complicated modular arguments. However,
it does require knowing a little basic group theory.

Fine... "more elementary" is a better choice of words. I agree entirely with what you say here and it was the idea I was trying to convey in my previous posting.

Originally Posted by R.D. Silverman

Please explain what you think "above first year calculus" really means.

High school mathematics and a year of calculus. In particular, no abstract algebra or number theory.

John

[Patrick123]

Mally:
I use Firefox for my web browser and it has the nifty feature <ctl> & + or - to incease or decrease the text size. I think MS Explorer has a similar feature.

Dario:
I must compliment you on the way you and the rest of the gentlemen involved in the mersennewiki have explained the concepts/definitions. Even a 'weekend mathematician' like myself can understand it.

Patrick

[xilman]

Originally Posted by R.D. Silverman

Please explain what you think "above first year calculus" really means.

I can tell you what I think it means, though whether that is relevant is an entirely different question.

I interpreted the word "above" as meaning "that which is typically studied later than".

That interpretation yields, to my mind, an entirely meaningful and sensible statement.


Paul

[Run800]

What I meant by "nothing above 1st year calculus level" is that calculus was the most recent and most advanced level of math I studied. I'm in my last year of high school, and unfortunately, calculus is the most advanced math my school offers.

If it's possible, I'd like a relatively short proof (less than 25 lines) that avoids using any math other than arithmetic, algebra I and II, geometry, and trigonometry.

However, if that is not possible, I am willing to learn number theory and/or abstract algebra that is needed in the proof. Just give me some good websites or books I could look into.

edit: Alpertron pointed to the Mersenne wiki. I've read it before, but I don't understand quadratic reciprocity.

[philmoore]

I originally posted the second part of the proof (sufficiency) before the first part (necessity) precisely because the sufficiency part of the proof did not require quadratic reciprocity which made it simpler, at least in terms of prerequisites, than the other part. Another editor thought the two pieces should be reversed, but unfortunately, as it is now written, it is rather confusing, as some constructions used in the sufficiency part are not explained until the necessity part. When I get time, I will try to re-edit it and make it clearer, but that will probably not be before spring break!

[philmoore]

Read the Law of Quadratic Reciprocity on the Mersennewiki - I think it summarizes the situation well. Actual proofs of the law can be somewhat subtle, however, but could be a good introduction into some nontrivial number theory.

[alpertron]

Originally Posted by philmoore

I originally posted the second part of the proof (sufficiency) before the first part (necessity) precisely because the sufficiency part of the proof did not require quadratic reciprocity which made it simpler, at least in terms of prerequisites, than the other part. Another editor thought the two pieces should be reversed, but unfortunately, as it is now written, it is rather confusing, as some constructions used in the sufficiency part are not explained until the necessity part. When I get time, I will try to re-edit it and make it clearer, but that will probably not be before spring break!

I was that wiki editor. If you think it is clear in the other way, just change it.

Please notice that the original version had several missing steps that I filled.

The ideal situation is that there should not be non-obvious steps. I prefer a larger proof where all steps can be understood.