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[popsofctown]

Any other opinion's on Lagrange's four square theorem?

[SirPeebles]

Wikipedia claims classification of finite groups has a thousands-of-pages proof.  Is that proof not universally accepted?

Eh.  There has been a lot of chatter that there could be an error somewhere, but no one has found one yet.  Most people think any mistake could be fixed if found.  Still, any theorem which relies on the classification theorem is usually advertised as such, as though there's an asterisk next to it.

The Lagrange four square thing looks really manageable.  Does it have unproven generalizations to talk about?

Lagrange's four square theorem is one of the earliest results in the subject of quadratic forms.  There are lots of unsolved problems, but I haven't looked at this stuff in years now and I'm totally blanking.  I know that it is related to the shortest vector problem, which as I mentioned in one of the go-to problems along with prime factorization used in cryptosystems.

[SirPeebles]

There is a theorem similar to Lagrange's Four Square Theorem, which states that if F is a finite field of odd order, then each polynomial with coefficients in F is a sum of three squared polynomials.

[popsofctown]

The hidden subgroup problem looked ok.  I'm not that great with groups though, I'm afraid I'll mess up.  I can't follow all of it.  I did our project where we proved that groups of order 4 are always Abelian pretty easily, but I had trouble understanding his discussion of subgroups, and the left cosets and right cosets involved in that.

[DG]

If you can find an example of using your number theory to solve problems outside the course itself, such as group theory to solve geometry problems, your tutor might be more impressed. Unfortunately I can't remember any of those applications. Perhaps they involved rotations. It was a long time ago for me.

[popsofctown]

I'd love to tie group theory to dominion somehow, but he's not that kinda guy and it isn't that kinda situation.

He likely would enjoy a geometry proof using group theory, but that might be beyond my skill.

[SirPeebles]

The hidden subgroup problem for just abelian groups is already sufficient for prime factorization, iirc.  I believe Shor's algorithm for factoring with a quantum computer is essentially just solving the hidden subgroup problem for abelian groups.  Quantum algorithms for the nonabelian case are unknown, but I think would solve many other problems.  But I may be mistaken.

[popsofctown]

Watno's RSA suggestion is the best so far, the only problem with it being that it is TOO good of a suggestion, so I might not be the only one who brings a baking soda volcano.  None of my peers have emailed me back and they probably won't.

[popsofctown]

The hidden subgroup problem for just abelian groups is already sufficient for prime factorization, iirc.  I believe Shor's algorithm for factoring with a quantum computer is essentially just solving the hidden subgroup problem for abelian groups.  Quantum algorithms for the nonabelian case are unknown, but I think would solve many other problems.  But I may be mistaken.

If I would have to get coset mud on my hands at all, I'll probably screw something up.  My mind isn't wrapped around it, so the risk of a false statement coming out of my mouth is drastically higher.

[SirPeebles]

I'd love to tie group theory to dominion somehow, but he's not that kinda guy and it isn't that kinda situation.

He likely would enjoy a geometry proof using group theory, but that might be beyond my skill.

There's a cute application of group theory to dice

http://en.wikipedia.org/wiki/Sicherman_dice

[popsofctown]

That looks really cool. I need to see about journal articles now.  Hopefully my university has something similar to my high school that can get me access to stuff.

[popsofctown]

Damnit.  I only found one source for Schicherman dice, and I ctrl+f'ed the journal article for the word "group" and got no hits.  Must be using a different approach.

[popsofctown]

Looking through cryptography results now.

Found a paper written totally in Spanish.  (Unseriously) considered translating it to English, like a confession bear on reddit said is a reasonably solid way to plagiarize.  Then remember my teacher is from Chile.  That would be a fail!  I have the academic integrity not to do that anyway though.

[popsofctown]

Browsing amongst the journals pertaining to "Cryptography" seems to be failing me, most of these are way out of my depth.  Our teacher, for the record, never taught us any cryptography at all

[SirPeebles]

Oops, it's not really group theory I guess.  I found my old algebra text, and it was an example of unique factorization for polynomials with integer coefficients.  Try looking up "Cyclotomic Polynomials and Nonstandard Dice" by Gallian and Rusin if you're interested.

Or just quit stalling and start working. :p

[popsofctown]

I'm honestly not stalling, I'm trying to find a suitable subject. 

The man is pretty hard to please.  He finds fault with every proof I ever do

[popsofctown]

I stumbled on Diffie-Hellman key exchange.  This looks really manageable and good.  Thoughts?

[SirPeebles]

That sounds good.  That's the application I had in mind when I suggested looking into the discrete log problem.  Plenty of unsolved work there.

[Kirian]

I'm glad Peebles was here with actual math knowledge... my understanding of anything beyond calculus comes from Wikipedia, Vi Hart, and xkcd.

[gman314]

I'm glad Peebles was here with actual math knowledge... my understanding of anything beyond calculus comes from Wikipedia, Vi Hart, and xkcd.

Discrete mathematics is not really "beyond calculus." The only reason it's perceived this way is because of the way many math curricula are set up, with calculus being the pinnacle of high school math. At least in Alberta, where I went to high school, they totally could have taught discrete math in grade 11-12 had they focused more on logic and proof, rather than on computation. The two are very different branches of mathematics, and there is no way that learning about continuous functions prepares you to learn about discrete mathematics. That being said, taking an introductory calculus course can prepare you for things like discrete math by starting to get you thinking about math in a more proof-focused manner than high school. Although, at my university, many first and second year calc courses miss proving almost entirely, leaving introductory linear algebra, ring theory and discrete math to introduce many students to proof.

[popsofctown]

Yeah, in my calc courses I didn't prove anything.

[gman314]

I took my school's honours calc courses in first year. They're completely proof-based, but pretty difficult. However, they are the reason that I'm miles ahead of a lot of people in my ring theory class and my "formal systems and logic in comp sci" course.

[popsofctown]

So it turns out, one of my classmates is doing Diffie Hellman, the other is doing something that has to do with Fermat.

Bad news is, that renders a good deal of reading I did on Diffie Hellman moot. 

Good news is, Watno's RSA suggestion seems available and fairly easy to work with.

[DStu]

"Every number can be represented as the sum of two primes" is unsolved.  But I think all the special cases for that are just empirical.

Every even.

>2.

If RSA is available, I would go for it.

[popsofctown]

Still working on this.

I've found my good source and started writing my paper, but my paper has made me realize I don't completely understand the concept.  This presentation could be an embarassment.

I don't understand how EulerFunction(n) helps you find e and d, which as far as I can tell are just two random numbers you picked because they are inverses of eachother. 
The system is based on congruency by the big number, but then they jump to the smaller number, Euler(n), and use that as a modulus instead for the interim stuff, and that gets me confused.  Is there a theorem I'm missing that says you can make that jump and it's basically the same thing?

The odds of me getting an answer in the next 3 hours is pretty low, bleh