number theory

This is 5th grade material, how hard could it be?

Integers Made up of Small Pieces Think back to elementary school, a time when things were simple and you had recess. Despite the most difficult math being times tables, we still said math classes were hard 💀 oh how times have changed.

Remainders and Division Before we get to a topic that is incredibly important in number theory, lets talk about division of integers and the different pieces that you can get when you divide numbers up into integer parts.

Unlimited Power ⛈️ Now that we have the machinery1 of the $\gcd$ and general divisibility, we can touch on a theorem that will unlock us the ability to prove a ton of problems that otherwise would have been super freaking annoying to solve without it.

The Math of Time In the Chapter on Set Theory, we discussed the idea of relations1 and specifically provided the example of modular equivalence on this page. In there we stated that the relation $$ a\equiv b \mod n $$ was true if both $a,b$ had the same remainder when divided by $n$.

Continuing Where We Left Off On this page we will discuss some theorems and topics that are slightly more advanced and would likely scare 👻 anyone away. Remember that equivalence $\mod n$ is literally an equivalence relation, that means we can actually create equivalence classes of items that are all equivalent to each other.

Euler’s Totient Function In this page we will discuss some interesting properties, and get us to the point that we need in order to understand RSA Encryption from scratch. Now that we have a distinction between residue classes and prime residue classes, we can see that some numbers seem to be “less prime” than other numbers.

What is RSA Encryption 📬 We’ve gone through a solid amount of number theory so far, and its crazy to think that this actually has a direct application in computer science, but it does.

Problem Description Let $P$ be a list of players that are queueing up to play a game of League of Legends, if you’re not a nerd feel free to switch this out to be some other team game like soccer or something.

Problem Description The Pythagorean Triples Coloring Problem1 takes in the set of all integers from $1\leq k\leq n$ for some value of $n$ and asks the question, is it possible to assign a color to every value in our set, such that no set of pythagorean triples is all the same color?