Many conjectures begin as simple patterns. After working out a few small cases, we start to hope that maybe the pattern holds in general. Many such conjectures have come from number theory and remain open to this day: take Twin Primes, Goldbach, or Collatz. Such intractible problems taint the hearts of many number theorists, leaving them hopeless that any simple conjecture has a simple proof.
The 15 Theorem, however, shines a beacon of hope onto the dark frontier of technical integral bounds and auxiliary terms in analytic number theory. It tells us about a large swathe of problems where a proof of the general result actually follows from checking a short list of small examples! In this talk, we introduce quadratic forms, discuss some general theory, and then outline the proof of the 15 Theorem.
DISCLAIMER: The 15 Theorem and its cousins, when proved in full detail, still use integral arguments. Optimizing the needed computations is actually a tremendous part of the legwork for proving any extension of the 15 Theorem! The arguments are super cool, but are also well beyond the scope of this talk. #AnalystsArePeopleToo
you can remove the hashtag if it's in bad taste i just want the analysts to know they're safe here even if their work scares me