Those primes and omega
On the 31st of October 2024, Mihai Prunescu gave a talk at the Logic Seminar in Bucharest presenting the results of our recent collaboration. This talk was entitled: “Primes Resurrection on Halloween”. Our collaboration pertains to an arithmetic term approach to the prime numbers. In the spirit of Halloween, I will leave it somewhat mysterious […]
On the n-th decimal digit of pi
Define the integer-valued function to return the -th decimal digit of . This is the sequence A000796 in the OEIS. We will now state a conjecture that is inspired by a limit appearing in an elementary proof of Stirling’s approximation by Jakub Smolík [1]. In his proof, Smolík shows: Conjecture. Let such that […]
An Easy Formula for Prime Numbers
A current research goal of mine is to find an elementary formula (arithmetic term) for the -th prime number. Today, I can at least report some partial progress. To provide some initial context: An arithmetic term is defined as an expression which uses only the elementary arithmetic operations: (*). (*) For a precise definition, please […]
Formulas for Euler’s Totient Function
I wish to share some intriguing formulas that I have recently discovered for Euler’s totient function, denoted as . Formulas I claim that these formulas are proved, though I omit the details for now. The formulas and their proofs will soon be added to a recent paper of mine: Elementary Formulas for Greatest Common Divisors […]
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 . […]