Orders
Orders modulo a prime, at the border of AIME and USA(J)MO but leaning a lot more towards the latter. Intended as an introduction into olympiad number theory.
Unit Overview
The main idea in this unit is to take the order modulo a prime $p$, defined as the least $m$ such that $a^m \equiv 1 \pmod p$. Another important concept is using the fact that there exists a primitive root modulo $p$, which is a value $a$ such that $\text{ord}_p(a)=p-1$. You are also introduced to Fermat's Christmas Theorem which is prevalent in many of the practice problems.
Notable Problems
- Fall OMO 2013/22: One of Evan's favorite examples which really tests how well you understand the notation.
- USA TST 2008/4: A difficult problem involving Fermat's Christmas theorem.
- HMMT November 2014/10: A classic orders example infused with a bit of casework.
- APMO 1997: A seemingly easy problem (but actually pretty hard) involving constructing a value.
Advice
This unit is a medium level B unit in terms of difficulty. It has some similarities to Prime Exponents with the idea of examining individual primes.