Mathematics

I am passionate about mathematics and am especially excited about Topology and Number Theory. Here are some expository papers I wrote recently in Number Theory and Combinatorial Game Theory.

Consecutive primes in arithmetic progressions

This is an expository paper I wrote in Spring 2024 (11th grade) for a course I took on Analytic Number Theory taught by Dr. Simon Rubinstein-Salzedo at Euler Circle. In this paper, I explain the proof of Shiu’s Theorem, a generalization of Dirichlet’s Theorem on primes in an arithmetic progression. The theorem states that in any arithmetic progression, there is a string of each length of consecutive primes. This can be proven by showing that there is a lower bound for the length of the longest string of consecutive primes in the arithmetic progression less than some x. The proof uses Maier’s matrix method and a version of the prime number theorem for arithmetic progressions.

Continued Fractions and Pell Equations

This is an expository paper I wrote in Fall 2023 (11th grade) for a course I took on Algebraic Number Theory taught by Dr. Simon Rubinstein-Salzedo at Euler Circle. In this paper, I describe properties of finite, infinite and periodic continued fractions and their convergents, and how they are used to determine the solutions to the Pell Equations. I discuss and prove some important results including Dirichlet’s Theorem on Diophantine Approximation and Euler and Lagrange’s Theorem that every quadratic irrational can be represented as a periodic continued fraction and vice versa.

A winning strategy in Fibonacci nim

This is an expository paper I wrote in Fall 2024 (12th grade) for a course I took on Combinatorial Game Theory taught by Dr. Simon Rubinstein-Salzedo at Euler Circle. In this paper, I explain the game of Fibonacci Nim with an example and define Zeckendorf representations, which are key to finding winning strategies. I then describe a winning strategy and prove that it works.