General note: Papers that are outside the scope of a mathematical research journal (e.g. a paper primarily concerned with physics, metaphysics, philosophy of mathematics, history of mathematics, mathematical education, or mathematical criticism) will almost certainly be rejected by any one of the journals above.
Editorial policy on submissions concerning famous problems
As JAMS editor, I receive a large number of submissions regarding either famous open problems (e.g. Riemann hypothesis, Goldbach conjecture, Navier-Stokes regularity, twin prime conjecture, etc.), or famous theorems (Fermat's last theorem, Four-color theorem, Cantor's theorem, Goedel's theorem, etc.). Such papersare held to an exceptionally high standard, and doubly so for a premier journal such as JAMS; extraordinary claims require extraordinary evidence, especially in view of the very many failed attempts to prove these types of problems (or disprove these theorems). In order to conserve limited refereeing resources, and to avoid possible embarrassment and damage to reputation for the submitter, I am thus imposing extremely strict quality standards on any such submission. In order to even be sent to a referee, any such submission must
Any submission which does not attempt to satisfy these requirements in good faith will be rejected without refereeing. All such decisions will be final. I will not consider any further revisions or resubmissions beyond the first when it comes to these sorts of submissions; it has to be perfect the first time, or it will not be considered at all.
Due to many existing time constraints, I will be unable to assist any prospective submitters with help in improving their mathematical exposition. If you are not a practicing research mathematician, my advice would be to first build up experience (and credibility) by working on less famous problems in the same area, in order to practice exposition skills, to learn basic techniques and literature, and to avoid common errors in the field. You may also wish to seek out a professional mathematician in your local area to collaborate or discuss mathematics with. See also my advice on writing papers, as well as my career advice page.