Achyut BharadwajDecember 2023
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.
...Achyut Bharadwaj, Lex Harie Pisco, Krittika Garg, Swayam Chaulagain,
Counsellor: Sanskar Agrawal,
Mentor: Nischay ReddyJune 2023
Introduction
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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.
...Achyut BharadwajApril 2023
Introduction–The Shuffling Problem
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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? In a previous article
we explored ways of trying to guess the outcome of a specific type of card
shuffling, known as the riffle shuffle. In this article, we modify the shuffling
algorithm to introduce a flip that introduces some interesting complexities.
...Achyut BharadwajMarch 2023
The Problem
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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?
...Achyut BharadwajSeptember 2022
Introduction
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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. In
this article I present an interesting way to characterize all such $n$
using concepts of finite fields.
...Achyut BharadwajSeptember 2022
Introduction
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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
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We first list down some basic definitions that will be used as we move
forward in this write-up.
...Achyut Bharadwaj, Tanmay Gupta, James Shuffelton, Toyesh Jayaswal, Matt BakerAugust 2022
Abstract
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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
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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. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
...Achyut BharadwajFebruary 2022
An Interesting Card Trick
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The Problem
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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?
...Achyut BharadwajSeptember 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 BharadwajSeptember 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
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Sending Messages
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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. In ASCII,
each character is given a value in binary.
“HELLO” is coded as: 1001000 1000101 1001100 1001100 1001111
So, Aditi has to send the above code to Bhaskar.
...