Students are building number theory from the definitions, discovering some "OP" results along the way.

 

I've learned a new word from my Multiplicative Number Theory students: OP. It literally stands for "overpowered," but it means something akin to "awesome" or "surprisingly cool." I'll use it in a sentence for you: "That proof is so OP!" It's a catch-all word that applies both when you realize that a proof is much easier than you thought, and also when a theorem you have proved is surprising. To take an example from this week, consider the totient function, which counts how many numbers less than or equal to n are relatively prime to n. It turns out, and this is such an OP fact, that if you sum up the totient function of the divisors of a natural number n, you get n.

This week, we worked on three different proofs of this amazing fact, digging into the "why." In the next week, we will contemplate this identity in the context of multiplicative functions to see that it is actually an example of a larger phenomenon. Our goal is to think deeply on several areas of number theory over the course of six weeks, and do a thorough job understanding these areas.



As we progress through different topics, I give out packets that serve as a framework for our exploration. The packets have numbered theorems in them and lots of blank space: students prove all of the theorems themselves, building number theory from the definitions. They have a lot to be proud of, halfway through the block. At this point we've proved 47 theorems and 5 lemmas, and used these to solve 28 exercises (not to mention the dozens of other contest math problems I have handed out as optional material.) Along the way, they've learned to be precise and give reasons for everything they write, answering the question "How do you know that?" until it meets our very high standards of rigor.

Our final goal for the class will be to understand when we can solve quadratic equations in modular arithmetic, and develop a proof of the law of quadratic reciprocity. Gauss, who proved this law, called it his "golden theorem" and said that it "must certainly be regarded as one of the most elegant of its type."  In modern day lingo, I would say that quadratic reciprocity is pretty OP.

--Sachi Hashimoto