Early view of tomorrow's video: Pi hiding in prime regularities

Beautiful video. Such simple techniques; and yet you reached a seemingly distinct result from where you started

Magnasium

Just wanted to say that I love the 60 fps!

Jonathan Fuzaro Alencar

Good catch, thanks! As to an analytic number theory book, I'd recommend the one by Montgomery and Vaughan

3blue1brown

Hey, great video! I came here to point out that, around 24:16, when you factor 45, you say "1 plus chi of 3" instead of "chi of 1 plus chi of 3" - the same for 5. When I was typing, I noticed they are the same, but that confused me long enough for me to come here. Also, I would like to ask if you have a recommendation for a book on analytic number theory? I'm taking a complex analysis class in college, and had a good real analysis and algebra classes.

I know what you mean about the "out-of-nowhere". I think one of the things I like so much about number theory is that it feels like we're only in the middle of cleaning it up. Since Fermat, there have always been things that we can prove and yet not understand more deeply, but in the last century-ish a lot of things are starting to fall together in ways that seem less arbitrary. Of all the proofs of quadratic reciprocity, for example, the only one I can ever remember is what comes out of class field theory, since it's the only one that doesn't feel like some sort of magic trick.

3blue1brown

Whoa, good catch! Very strange how it's there, but hidden, since it pops into existence when they are organized into columns.

3blue1brown

Just finished. Had to read the wiki on why 1 mod 4 is sum-of-two-squares but 3 mod 4 isn't. Good stuff. I'm probably just saying this because analysis is my field, but number theory stuff always seems so out-of-nowhere. It does, however, embody the "playfulness" toward math that you're trying to encourage. For me, math is more down-to-business. I've got this thing I want to estimate, so I'm going to break it into parts A, B, C, D, and proceed to hit each part with a baseball bat until it behaves.

Jacob Mirra

It doesn't bother me in the least, but I could imagine that your saying, "you're screwed" at 23:20 might bother some listeners. How about "you're out of luck" instead.

Steve Muench

<a href="https://twitter.com/3Blue1Brown/status/804168776761503744" rel="nofollow noopener" target="_blank">https://twitter.com/3Blue1Brown/status/804168776761503744</a>

15:41 you forgot a factor.

Jacob Mirra

Grant, did you mean to say, "...named after Martin Sheen?" instead of Gauss?

Steve Muench