Open Problems in Number Theory?

Question

Any one know of any good (by which I mean not the Riemann Hypothesis or the Twin Prime Conjecture) open problems in number theory, particularly algebraic number theory?

Answer 1

This field is not really designed for outsiders. I am not sure this is the right place to ask.

Actually I have a question but I don't know if that's the kind of problem you have in mind. It comes from geometry and reads like this.

For which quadruples of integers (p,q,r,s) is
(ps-qr)^2 (qs-pr)^2 + pqrs (p+q)^2 (r+s)^2 a perfect square?

I would like a representation like for Pythagoreans triples.

Answer 2

As a grad student, you probably know these already but, just in case you still haven't heard about the Milenium Problems contest, here they go:
* Birch and Swinnerton-Dyer Conjecture
* Hodge Conjecture
* Navier-Stokes Equations
* P vs NP
* Poincaré Conjecture
* Riemann Hypothesis
* Yang-Mills Theory

I believe the Poincaré Conjecture has been solved this year, though.

PS: Good luck. :-)

Answer 3

Fermat's Last Theorem was solved not too long ago, but it required over two hundred pages of modern mathematics. When Fermat first developed the theorem, he stated that the proof was simple, but he could not write it in the margins. There may still be a simple solution to the theorem that no one has yet to discover.

Fermat's Last Theorem: a^n + b^n = c^n where a, b, c and n are integers does not hold true for any n greater than 2.

Answer 4

Intuitively, I think Xi's should be the set of smallest distinct primes x1=2 x2=3 x3=5 x4=7 this (log time) algorithm should give the best answer by a factor of 1/2