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 ---------------- 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/Exponentiation_by_squaring https://en.wikipedia.org/wiki/Modular_exponentiation 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%27s_identity https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm 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.2016.11.019 Hensel's lemma gives conditions for Newton's method to work in the p-adic numbers: https://en.wikipedia.org/wiki/Hensel%27s_lemma ---------------- 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? ---------------- 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: https://twitter.com/ericrowland