Month: November 2024

Theoretical Background Finite Fields and Irreducible Polynomials A finite field $\mathbb F_p$ consists of a finite set of elements with well-defined addition and multiplication operations, where $p$ is a prime number. A polynomial $f(x) \in \mathbb{F}_p[x]$ is irreducible over $ \mathbb{F}_p $ if it cannot be factored into the product of two non-constant polynomials in… Continue reading The Cubic Frobenius Primality Test by ChatGPT

It’s a test that any prime number passes, but that only a few composite numbers pass. The test is made up of three congruences that are proved using basic Galois Theory. The setup is an irreducible polynomial $f$ over $\mathbb F_{p}$ with $f = X^3 – rX – s$, and $r>0$ $s>0$. Additionally, we require… Continue reading What is a cubic Frobenius primality test?

\section*{Progress Update: Scaling Prime-Counting to 100 Billion} I’m excited to share some recent progress on my prime-counting program. After numerous iterations and optimizations, the program is now fully operational and demonstrating impressive performance metrics. \subsection*{Current Performance} The program efficiently processes large integer ranges with the following benchmarks: \begin{itemize}\item 1 Billion Integers: Approximately 6 minutes.\item 100… Continue reading Update on the cubic Frobenius primality test

Let $f$ be the polynomial $X^3 -rX -s$, where $r>0$ and $s>0$ and let $\Delta$ be its discriminant. Suppose that $\Delta$ is a square, and that $p$ and $q$ are primes such that $\Delta$ divides $p-q$. Under those conditions, is it always the case that $f$ is irreducible in $F_{p}$ iff f is irreducible in… Continue reading A question on polynomial irreducibility over finite fields

Let $f = X^3 – 3X – 1$ and let $p$ be a prime such that $f$ is irreducible over $F_{p}$. Let $L$ be a splitting field of $f$ over $F_{p}$. For $k>=0$, let $V_{k} = \alpha^k + \beta^k + \gamma^k$ , where $alpha$, $beta$ and $gamma$ are the three roots of $f$ over $L$.… Continue reading Question in Galois Theory and Finite Fields

Certainly! Here’s a comprehensive blog post on the Cubic Frobenius Primality Test based on the polynomial ( f(X) = X^3 – 3X – 1 ). This post is structured to be engaging and informative, suitable for readers with a keen interest in number theory and computational mathematics. Introducing the Cubic Frobenius Primality Test Primality testing… Continue reading Another cubic Frobenius test based on $X^3-3X-1$

This memo explores the properties and distribution of prime numbers representable by the quadratic form $2x^2 + xy + 3y^2$. Utilizing an optimized primality testing method based on matrix exponentiation, we efficiently identify such primes, even extending to 300-digit numbers. The approach hinges on the irreducibility of the polynomial $X^3 – X – 1$ modulo… Continue reading On Primes of the Form $2x^2 + xy + 3y^2$

Certainly! Below is the reformatted version of the memorandum tailored for your WordPress blog. This version replaces the \[ … \] delimiters with $$ … $$ for display math, ensuring compatibility with your MathJax plugin. Inline mathematical expressions remain enclosed within single $ … $ as before. Enhancing the Frobenius Primality Test Using Cubic Recurrence… Continue reading Last Memo, with improved Latex rendering

Certainly! Let’s incorporate your insightful feedback and the confirmation from PARI/gp to finalize the derivation of ( V_{p+2} ) in the context of the Cubic Frobenius Primality Test. This will ensure that our framework is both accurate and robust. Below is the updated and corrected derivation, formatted in LaTeX for seamless integration into your WordPress… Continue reading Memorandum for the record: cubic Frobenius test, a work in progress

Certainly! Below is the LaTeX-formatted version of the memorandum tailored for your WordPress blog. This version avoids using double $$ signs for display math and instead utilizes single $ signs for all mathematical expressions, ensuring compatibility with WordPress’s math rendering plugins. Enhancing the Frobenius Primality Test Using Cubic Recurrence Relations Introduction In the pursuit of… Continue reading Converted to Latex (about Symmetric polynomials,Galois theory and primality testing)