*Achyut Bharadwaj, Lex Harie Pisco, Krittika Garg, Swayam Chaulagain,
Counsellor: Sanskar Agrawal,
Mentor: Nischay Reddy*##### June 2023

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.
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*Achyut Bharadwaj*##### April 2023

Introduction–The Shuffling Problem # Card shuffling is an important part of playing any card game. When a card deck isn’t shuffled properly, it leads to uneven and unfair distribution of cards. It would certainly help if you knew what cards other players had!
Suppose we have a specified shuffling algorithm. Is it possible for us to guess the outcome of the card shuffling? Is it possible that the card deck will at some point return to its original configuration?
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*Achyut Bharadwaj*##### March 2023

The Problem # Take a square piece of paper. Take a toothpick of a given length. At intervals equal to the length of the toothpick, draw lines on the piece of paper. Now, randomly toss a bunch of such toothpicks so that they fall over the paper. What fraction of the toothpicks will fall in a way so that they intersect one of the lines drawn? In other words, what is the probability of a single toothpick falling over a line?
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*Achyut Bharadwaj*##### September 2022

Introduction # Consider a prime $p$. For what integers $n$ does $p$ divide all of $$\binom{n}{1}, \binom n 2, \binom n 3, \dots, \binom{n}{n-1}?$$ Can we characterize all such $n$ given a value of $p$? It turns out that this happens if and only if $n$ is a perfect power of $p$. How do we prove this? In fact, there exists a simple proof using elementary methods. But as always, it is both fun as well as good to prove everything twice.
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*Achyut Bharadwaj*##### September 2022

Introduction # A well known theorem about finite fields states the following.
$F$ is a finite field if and only if $|F| = p^k$ for some prime $p$ and positive integer $k$.
In this write-up, we prove the first part of the above theorem, i.e. if $F$ is a finite field, then $|F| = p^k$ using a different approach.
Introduction to Finite Fields # We first list down some basic definitions that will be used as we move forward in this write-up.
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*Achyut Bharadwaj, Tanmay Gupta, James Shuffelton, Toyesh Jayaswal, Matt Baker*##### August 2022

Abstract # The problem of balancing centrifuges is equivalent to finding sets of $k$ $n$-th roots of unity that sum to $0$. We can represent each slot in the centrifuge as an $n$-th root of unity as the slots in the centrifuge are also equally spaced and equidistant from the origin.
Acknowledgements # Accepted for publication at the Joint Mathematics Meetings, Boston, January 4 – 7, 2023.
This work was carried out as part of a research project at PROMYS-2022.
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*Achyut Bharadwaj*##### February 2022

An Interesting Card Trick # The Problem # Suppose you have a deck of $52$ cards. You perform a riffle shuffle on these cards. Can you say what the new position of the first card in the deck is, after the shuffle? What about the second card in the deck? The third? In general, is it possible to predict the new position of the $k$th card in the original deck after a new deck is produced through a riffle shuffle?
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*Achyut Bharadwaj*##### September 2021

cryptography,

RSA,

encryption,

decryption,

one-way function,

factorization,

prime,

modular arithmetic,

modular inverse,

kid-RSA,

Euclid's division,

gcd,

Bezout,

Euler,

totient,

Fermat,

primality test,

Miller-Rabin A presentation on the mathematics of cryptography, with an emphasis on RSA. A link to the writeup is here.

*Achyut Bharadwaj*##### September 2021

cryptography,

RSA,

encryption,

decryption,

one-way function,

factorization,

prime,

modular arithmetic,

modular inverse,

kid-RSA,

Euclid's division,

gcd,

Bezout,

Euler,

totient,

Fermat,

primality test,

Miller-Rabin A presentation on the mathematics of cryptography, with an emphasis on RSA. A link to the slides is here.
Basic Cryptography # Sending Messages # Say, there are 3 people, Aditi, Bhaskar and Diti.
Aditi wants to send the message “HELLO” to Bhaskar, using her computer.
Computers, however, can only store numbers. How will she send the message “HELLO” to Bhaskar?
To store characters, there is a code called the ASCII code.
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*Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny Li*##### August 2021

Acknowledgements # This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
Definition and Preliminaries Notation # Let us define a polynomial $f(x)$ to be $$f(x) = \sum_{i = 0}^n a_i x^i$$ for some $a_i \in \mathbb{C}$, for all $0 \le i \le n$ and $n \in \mathbb{N}_0$. We define the degree of a polynomial to be the largest non-negative integer $n$ such that $a_n \not= 0$ and let degree of the polynomial $f(x) = 0$ be $0$.
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