Month: May 2024
Multivariate Binary Representation
In this blog post, I introduce Multivariate Binary Representation: A method for encoding the binary expansions of integers as multivariate monomials using a polynomial quotient ring. This encoding process involves modular exponentiation within a specific ring structure, referred to as , to uniquely represent any positive integer . Let . Consider the recursive polynomial quotient […]
Strange Formulas for Binomial Coefficients
Binomial coefficients have long been an obsession of mine, since my earliest days of studying the integers, when I naively rediscovered Pascal’s triangle through simple counting. These unassuming numbers, often written as , have a surprising depth and elegance that never cease to amaze me. For those who may be unfamiliar, binomial coefficients are the […]
A Conjecture on Primality Testing
We begin with a conjecture on primality testing, which I first shared on my personal GitHub early last month. Conjecture 1. Let such that . Then, the following polynomial congruence holds iff is prime: By the binomial theorem, it is easy to see that the polynomial congruence will hold for all prime . […]