I saw a curious identity and I got curious about more factorizations of this type: . I did a computer search and found an infinity family . The original one is an except and every other one seems to fit into that family. I don't have a proof though.
An interesting problem I saw on MSE was that if is a polynomial who always takes on square values, can we write for some polynomial ? The answer is yes and this is proved in a paper here: http://www.mast.queensu.ca/~murty/poly2.pdf
The author Ram Murty has a book about transcendental numbers which looks very interesting. I remember I was working on Davenport and that was going well - I took a digression to study quadratic forms a little. I would definitey like to continue working on and learning about QFs.
Something that has been nagging at me for a while is the Newton Polygon. It's what lies behind Eisensteins criterion and that lemma is so "magic". I would like to understand it deeper and I've heard it's explained by the Newton Polygon but I don't have a good reference for learning about that.
I wrote the first part of this post a few days ago. Turns out the problem is actually open!
I'm just trying to think about what it is I should be doing with my time. I miss programming so much but I can't spend time doing that or it hurts my arms. The blog did help me find my feet a bit with math, I should continue studying mathematics.
I was wondering if I should learn Hartshorne, but it's so much work and it's all about instead of . That reminds me: I read a comment that homogeneous equations are curves but non-homog. are surfaces. We have more tools for dealing with curves.
There was a very impressive resolution of a quintic diophantine equation here: http://mathoverflow.net/questions/224232/rational-points-on-the-quintic-circle-x5-y5-7
I think I will change my blog software to use mathjax too. I really like katex but it seems to require javascript and it has a very low feature set. Hopefully in the future I can change back.