Heavy NT
Heavy machinery in number theory: Vieta jumping, quadratic reciprocity, and some big-name theorems you may or may not have heard of.
This unit covers a variety of high powered theorems and concepts found in both computational and olympiad settings.
Included are tricks to finding integer or rational families of solutions on strange curves using the theory of Pell equations, Vieta Jumping, and a brief touch at elliptic curves in the walkthrough.
Also in this unit are results relating to the behavior and distribution of primes: Quadratic Reciprocity, Zsigmondy theorem, Bertrand postulate, and Dirichlet are mentioned.
Heavy NT can be quite tough. It could be considered an extension of Expon NT in terms of theory, and is also similar to Prime Exponents and Orders as careful analysis mod p is key to many problems.
Notable Problems
2022 AMC12A 16: Using Pell equation theory.
QR 7 mod 8: Elementary and classic application of Quadratic Reciprocity.
2020 HMMT T9: A nice example showing the power of Quadratic Reciprocity.
2013 Iran TST: A beautiful example of using Quadratic Reciprocity to derive contradictions.
2003 IMO 2: A classic instance where Vieta Jumping comes in handy.
2014 TST 2: Difficult problem using Legendre symbols.