$p$-adic numbers

Ikshvaku and the 163-adic Udupi Shri Krishna Bhavan

Achyut Bharadwaj
December 2023
Udupi, Krishna Bhavan, Ikshvaku, 163-adic, $p$-adic numbers, UChicago, University of Chicago, Masala Dosa, Masala Dosa Infinity, Olive Garden

This is a short story that I initially wrote for my UChicago supplemental essay for the following prompt:

“If there’s a limited amount of matter in the universe, how can Olive Garden (along with other restaurants and their concepts of food infinity) offer truly unlimited soup, salad, and breadsticks? Explain this using any method of analysis you wish—physics, biology, economics, history, theology… the options, as you can tell, are endless.” – Inspired by Yoonseo Lee, Class of 2023.

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The $p$-adic Number System and the Artin-Hasse Exponential

Achyut Bharadwaj, Lex Harie Pisco, Krittika Garg, Swayam Chaulagain, Counsellor: Sanskar Agrawal, Mentor: Nischay Reddy
June 2023
$p$-adic numbers, Artin-Hasse exponential, radius of convergence, power series, integral power series, discs, leftist numbers

Introduction #

In this paper, we explore the $p$-adic system, by defining it in multiple ways: as an extension of the $p$-adic integers, as well as an extension of the rationals. We then proceed to perform analysis in the $p$-adics, by defining convergence, continuity and discs. We then describe exponentiation and logarithmic functions over the $p$-adics as functions derived from power series. We explore the radius of convergence and other properties of these functions. We then explore the Artin-Hasse exponential, which, though seemingly random, turns out to be an integral power series.

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