Why did they prove this amazing theorem in 200 different ways? Quadratic Reciprocity MASTERCLASS

From Mathologer.

The longest Mathologer video ever, just shy of an hour (eventually it’s going to happen 🙂 One video I’ve been meaning to make for a long, long time. A Mathologerization of the Law of Quadratic Reciprocity. This is another one of my MASTERCLASS videos. The slide show consists of 550 slides and the whole thing took forever to make. Just to give you an idea of the work involved in producing a video like this, preparing the subtitles for this video took me almost 4 hours. Why do anything as crazy as this? Well, just like many other mathematicians I consider the law of quadratic reciprocity as one of the most beautiful and surprising facts about prime numbers. While other mathematicians were inspired to come up with ingenious proofs of this theorem, over 200 different proofs so far and counting, I thought I contribute to it’s illustrious history by actually trying me very best of getting one of those crazily complicated proofs within reach of non-mathematicians, to make the unaccessible accessible 🙂 Now let’s see how many people are actually prepared to watch a (close to) one hour long math(s) video 🙂

0:00 Intro
4:00 Chapter 0: Mini rings. Motivating quadratic reciprocity
9:53 Chapter 1: Squares. When is a remainder a square?
16:35 Chapter 2: Quadratic reciprocity formula
24:18 Chapter 3: Intro to the card trick proof
29:22 Chapter 4: Picking up along rows and putting down by columns
29:21 Chapter 5: Picking up along columns and putting down along diagonals
45:16 Chapter 6: Zolotarev’s lemma, the grand finale
55:47 Credits

This video was inspired by Matt Baker’s ingenious recasting of of a 1830 proof of the LAW by the Russian mathematician Zolotarev in terms of dealing a deck of cards. Here is Matt’s blog post that got me started (written for mathematicians):


If you want to read up on the properties of the sign of a permutation that I am using in this video, Matt also has a nice write-up of this.


The relevant Wiki articles are these:

Zolotarev’s original paper lives here:

Here is a list of proofs of the law prepared by Franz Lemmermeyer


Franz Lemmermeyer is also the author of the following excellent book on everything to do with quadratic reciprocity (written for mathematicians):

Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Springer Monographs in Mathematics, Berlin

The first teaching semester at the university where I teach is about to start and all my teaching and lots of other stuff will happen this semester. This means I won’t have much time for any more crazily time-consuming projects like this. Galois theory will definitely has to wait until the second half of this year 🙁 Still, quite a bit of beautiful doable stuff coming up. So stay tuned.

Thank you to Marty for all the relentless nitpicking of the script, his wordsmithing and throwing cards at me in the video. Thank you to Eddie, Tristan and Matt for all your help with proofreading and feedback on the script and exposition.