Category: History

Date: June 26, 2025 Subject: Conclusion of Overlap Search Between Fermat Base 2 Pseudoprimes and Cubic Primality Test

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)

Preface In 1858, Richard Dedekind faced the challenge of teaching differential calculus without a rigorous foundation in arithmetic. Dissatisfied with relying on geometric intuition, he sought a purely arithmetic basis for the principles of infinitesimal analysis. This led him to the concept of continuity and the creation of irrational numbers, aiming to provide a scientific… Continue reading Summary of “Stetigkeit und irrationale Zahlen” by Richard Dedekind

by ChatGPT-4o The recent announcement by MUFON regarding the UAP materials case from Russia presents compelling evidence supporting the extraterrestrial hypothesis. The materials in question, received by a Russian UFO investigator, have undergone rigorous testing by both Russian and U.S. labs, confirming that 90% of the sample consists of unknown substances. This defies conventional scientific… Continue reading MUFON Materials Advocacy

Hi all, I’ve just read the testimonial given by Robert Lazar who claims that he was amember of a compartmentalized team of researchers that were reverse engineeringan extraterrestrial vehicle at site S-4 inside Area 51 within the Nevada TestSite. My skeptical foundations are beginning to crumble, please debunk ! I’ve heard of strange rumours before,… Continue reading Sci.skeptic USENET post of Jan 5 1994 Re: Bob Lazar

The “Wilson Notes,” also known as the “Admiral Wilson UFO Documents,” refer to a series of controversial and purportedly leaked notes detailing a conversation between Dr. Eric Davis and Admiral Thomas R. Wilson in 2002. The notes describe Wilson’s alleged discovery of a highly secretive black program focused on reverse engineering alien technology, which he… Continue reading ChatGPT-4o analysis of the Wilson Notes

The Nakba (Arabic: النكبة, “the catastrophe”) refers to the 1948 Palestinian exodus, a significant event during the Arab-Israeli conflict, where over 750,000 Palestinian Arabs were expelled or fled from their homes in Mandatory Palestine. This exodus occurred during and after the 1948 Palestine war, leading to the destruction of Palestinian society and culture. The Nakba… Continue reading Summary of Wikipedia article on the Nakba by ChatGPT-4o

I’m continuing my evaluation of Khashin’s Frobenius Primality Test from his 2013 preprint at the URL https://arxiv.org/abs/1307.7920. To make it easier to assess the accuracy of this test, I assume that in a first round, one conducts a Fermat base 2 test, which will detect all composites that are not base 2 pseudoprimes. In round… Continue reading Continued Evaluation of a Frobenius Primality Test

With respect to the enhanced Lucas test for n with parameters P and Q, using congruences for both of the Lucas sequences U_n and V_n,I’ve come to the realization that the Lucas congruences are equivalent to a specific test inspired by the Frobenius endomorphism in the ring Z/nZ[sqrt(D)] where D=P^2-4Q.Specifically, and assuming P and Q… Continue reading Equivalence of Enhanced Lucas Test with a Frobenius Test

Certainly! Here’s a concise summary of the equivalence between an enhanced Lucas test and a Frobenius test: The equivalence between an enhanced Lucas test, which includes congruences for both (U_n) and (V_n) sequences, and a Frobenius test, which checks if (x^n = \text{conjugate}(x)) for a specific choice of (x), lies in their shared foundation in… Continue reading Enhanced Lucas Test, and a Frobenius Test by Chat-GPT

The exploration of primality tests is a foundational pillar in the realm of computational number theory, with wide-ranging applications from cryptography to the distribution of prime numbers. Among the myriad of tests developed over the years, the Lucas sequences and Frobenius tests stand out for their unique approaches and theoretical underpinnings. Our recent investigation has… Continue reading Exploring a Lucas-Frobenius Tests Connection by Chat-GPT

A promising primality test is Sergei Khashin’s Frobenius test, as described in a 2013 arxiv preprint of his, Counterexamples for Frobenius primality test, available from the url https://arxiv.org/abs/1307.7920. The idea of the Frobenius test originated with Jon Grantham, see for example his 1998 article A Probable Prime Test With High Confidence available from the url… Continue reading Evaluation of a Frobenius Primality Test Using Pseudoprimes Tables

Given an odd number n>=3, non-square and an odd number c such that Jacobi(c,n)=-1 and 1<c<n/2,Khashin’s Frobenius primality test checks whether the congruence (1+sqrt(c))^n == 1-sqrt(c) (modulo n)holds. This congruence holds whenever n is a prime number, thanks to properties of the Frobeniusautomorphism map x |-> x^p where p is prime and x is in… Continue reading False positive rates in Khashin’s Frobenius primality test

I’ve been experimenting with S. Khashin’s Frobenius primality test, as described in a preprint of his, Counterexamples for Frobenius primality test, at https://arxiv.org/abs/1307.7920 . Given an odd number n>1 that is not a square, let c be an odd prime number with (c/n)=-1, (c/n) being the Jacobi symbol. If n is prime, then (1+sqrt(c))^n ==… Continue reading The Frobenius Test Finds No Liars Below 2000 Among the First Million Test Numbers

The Frobenius primality test is an algebraic type primality test which perhaps deserves to be better known. Like other tests, it is sure to label genuine prime numbers as prime, but is liable to misidentify some composite numbers as primes. The test was first described by Jon Grantham in a 1998 preprint. There are a… Continue reading Prime Deception: Counting Liars for the Fermat, Miller-Rabin and Frobenius Primality Tests