1 Billion is Tiny in an Alternate Universe: Introduction to p-adic Numbers

The p-adic numbers are bizarre alternative number systems that are extremely useful in number theory. They arise by changing our notion of what it means for a number to be large. As a real number, 1 billion is huge. But as a 10-adic number, it is tiny! #SoME2

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Notes and references:

The last 30 digits of 2^1000000 and other large powers can be computed using modular arithmetic, by working modulo 10^30. In Mathematica, use the function PowerMod. In Python, use the third argument of pow. These functions implement the method of repeated squaring or one of its variants:
https://en.wikipedia.org/wiki/Exponen...
https://en.wikipedia.org/wiki/Modular...

Bézout's identity can be used to prove that the numbers from 2 to p-2 pair up perfectly, and the partner of a given number can be computed using the extended Euclidean algorithm:
https://en.wikipedia.org/wiki/Bézout%...
https://en.wikipedia.org/wiki/Extende...

The 2-adic limits arising from the (2^n)th Fibonacci numbers were established on page 216 of this paper:
Eric Rowland and Reem Yassawi, p-adic asymptotic properties of constant-recursive sequences, Indagationes Mathematicae 28 (2017) 205–220.
https://doi.org/10.1016/j.indag.2016....

Hensel's lemma gives conditions for Newton's method to work in the p-adic numbers:
https://en.wikipedia.org/wiki/Hensel%...

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0:00 Introduction
2:16 Properties of the real numbers
3:19 10-adic integers
6:55 Properties of the 10-adic integers
10:06 Division?
12:47 Limit points
13:50 5-adic limit
15:36 Fibonacci numbers
16:31 Square roots of -1
18:25 What are p-adics good for?

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Animated with Manim. https://www.manim.community
Music by Marc Rowland and Cody Leavitt.
Thanks to @catpfaff for helpful feedback on an earlier version.

Web site: https://ericrowland.github.io
Twitter:   / ericrowland