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

I’ve collected in one spot the functions for using Sergey Khashin’s Frobenius primality test. The idea of a Frobenius test is due to Jon Grantham.

I’m posting this so I can refer to it later. The number being tested is n=(82065^19937 -1)/(82065-1). This is one part of a Miller-Rabin strong primality test. One checks that diff is -1, then one replaces the 32 on line 3 of the script by 16, and the value of res ought to be 1.… Continue reading An OpenPFGW script that does a partial Miller-Rabin test on a large number

Given an odd number N with k bits where k>=2, we can ask for the nearest k-bit prime p in Hamming distance, that is the number of bit positions where p and N in base 2 differ. Up to 2×10^9, I get distances of 2 or less. Therefore, among the set P_N = { primes… Continue reading Odd numbers below 2e9 and close primes in Hamming distance (Updated)

The golden ratio φ is an irrational number, approximately equal to 1.61803, and it’s known for its unique and pleasing properties in mathematics, art, and nature. The Lucas numbers are an integer sequence similar to the Fibonacci numbers and are defined by the recurrence relation (L_n = L_{n-1} + L_{n-2}), with initial terms (L_0 =… Continue reading About phi^n, the nth power of the golden ratio

In the past week, I’ve run an experiment with ChatGPT to answer the question: Is it feasible to use Chatbots to ease and accelerate the writing of computer code to search for answers to very hard combinatorial problems? From my experience so far over five days, I’m inclined to say yes. The problem I decided… Continue reading Chatbots as Programming Assistants

Over the past 24 hours, we’ve made significant progress in developing and optimizing an algorithm to find large Salem-Spencer sets within a given range. Here’s a recap of the key developments: As a result of these efforts, the program has successfully found Salem-Spencer sets of significant sizes, with the largest found being a set of… Continue reading ChatGPT memo concerning programs to find Salem-Spencer sets

[latexpage] It seems like you are exploring the relationship between ( f(u) ) and the non-trivial zeros of the Riemann zeta function, specifically focusing on how the residual term ( \text{Residual}_T(u) ) in your function behaves as ( T ) approaches infinity, particularly in the context of worst-case scenarios for ( u ). The function… Continue reading test 3

It seems like you are exploring the relationship between ( f(u) ) and the non-trivial zeros of the Riemann zeta function, specifically focusing on how the residual term ( \text{Residual}_T(u) ) in your function behaves as ( T ) approaches infinity, particularly in the context of worst-case scenarios for ( u ). The function (… Continue reading test 2

[latexpage] ME: Suppose we write f(u) = -sum_{rho} exp(iIm(rho)u)/rho = S_{T}(u) + Residual_T (u), how does the Residual_T(u) behave as T -> oo in worst-case u ? ChatGPT’s answer: It seems like you are exploring the relationship between ( f(u) ) and the non-trivial zeros of the Riemann zeta function, specifically focusing on how the… Continue reading ChatGPT-4 answers a math question of mine

What follows is ChatGPT’s response to the text of three tweets I made: Your analysis of the limitations inherent in using the von Mangoldt explicit formula for ( \psi ) is insightful, particularly in terms of capturing high-frequency oscillations in ( \psi(e^u) – e^u ). The von Mangoldt formula indeed sums contributions from non-trivial zeta… Continue reading Short-term prime anomalies and Skewes number

Objective: This memo encapsulates the empirical findings and insights gained since the commencement of the Skewes Challenge, focusing on the behavior of the function ( f(u) = \frac{\psi(e^u) – e^u}{\sqrt{e^u}} ) and the comparison of arithmetic and analytic methods in identifying ( f(u) )-champions. 1. Overview of the Skewes Challenge: 2. Methodological Approach: 3. Key… Continue reading Memo: Summary of Empirical Findings – Skewes Challenge – January 5 2024

Today, I’d like to propose an intriguing mathematical challenge: the Skewes Challenge. It’s a deep dive into the world of prime numbers, inviting mathematicians and enthusiasts to explore up to the significant scale of 10^316. What is the Skewes Challenge?The challenge focuses on finding a number m where f(m) > C, with f(m) = (psi(m)… Continue reading Introducing the Skewes Challenge: Exploring Prime Numbers Up to 10^316

Introduction:This memo provides an overview of our ongoing research into a specific set of numbers associated with significant deviations in prime-related functions. These numbers, tentatively referred to as “numbers of type Skewes,” represent points where the normalized deviation of the Chebyshev psi function (( \psi(x) )) from its expected value ( x ) reaches new… Continue reading Ongoing Research on Record-Breaking Numbers in Prime Distribution (Numbers of Type Skewes)

Certainly! Here’s a 300-word memo outlining your plan: Memo: Analysis Plan for Investigating Correlations in Prime Number Distributions Objective:To explore the correlation between the arithmetic function (\psi) and the independent variable (J) in the context of prime number distributions. Methodology: Expected Outcome:The analysis is expected to reveal a substantial number of statistically significant p-values, indicating… Continue reading Memo from ChatGPT-4 on plan for project

Certainly, here’s a memo discussing the significance of increasing ( T ) in Richard Sherman Lehman’s function ( S_T(u) ) and its implications for understanding prime number counts: Memo: Implications of Increasing ( T ) in Lehman’s Function ( S_T(u) ) for Prime Number Counts Background:Richard Sherman Lehman’s function ( S_T(u) ) is a summation… Continue reading Lehman’s function S_T(u)

Exploration of the primes goes back to antiquity. Recently, a new dimension has emerged in this exploration, linking the distribution of prime numbers with the phenomenon of 1/f noise. This connection, while nascent in its exploration, opens a window to a deeper understanding of primes through the lens of complex systems theory. Commonly encountered in… Continue reading Prime Numbers and 1/f Noise

Certainly! Here’s a memo on the presence of pink noise in prime number distribution, crafted in a serious tone: To: [Recipient] From: [Your Name] Date: [Date] Subject: Emergence of Pink Noise Patterns in Prime Number Distribution Introduction Recent analytical explorations into the distribution of prime numbers have unveiled a fascinating phenomenon: the presence of pink… Continue reading Pink noise in prime counts? (collaboration w/ Chat-GPT)

Memo: Progress and Future Directions in c(N_k) Computations Date: [Current Date] Subject: Overview of Recent Work and Prospective Steps in c(N_k) Computations Overview: Recent efforts have focused on computational analysis of Nicolas constants ( c(N_k) ) for large primorials. Utilizing the Primesieve library, substantial computations have been conducted, particularly with values up to ( M… Continue reading Memo for the record by Chat GPT 4 on some number theory computations

(Q1) I don’t think Rayo’s Number is well-defined. What do you think? Rayo’s number is an example of a very large number defined in a highly abstract and complex way. It was created by Agustín Rayo in 2007 as part of a “big number duel” against Adam Elga. The formal definition of Rayo’s number involves… Continue reading A discussion with Chat GPT-4 on Rayo’s Number

The opening was a French Defense, Winawer, Poisoned Pawn Variation with 7… Nf5. Although 7… Nf5 is a book move, it could still be a mistake. At depth=71, Stockfish 16’s choice is 7… Kf8. Next, I might try the French Defense, Winawer, Poisoned Pawn with 7… Kf8. The game is copied below. The game with… Continue reading Stockfish 16 depth=63 vs Stockfish 15.1 depth=49 (1-0)

The opening was the French Defense, Winawer, Poisoned Pawn Variation (with 7. Qg4). It ended up in a rook and pawn endgame. I’m wondering if Black playing 7… Nf5 could be a mistake. Although 7… Nf5 is a book move, I suppose it could still be a mistake. Next, I might try Stockfish 16 depth=59… Continue reading Stockfish 16 depth=59 vs Stockfish 15.1 depth=47 (1-0)

This was a French Defense Winawer Variation game. White played 7. Qg4: the Poisoned Pawn Variation. It would be interesting to have more games in this variation (Winawer Poisoned Pawn). The game is at the link: https://www.chess.com/analysis/game/pgn/B21fFYjev?tab=analysis .

White played the Sveshnikov Variation of the Sicilian Defense. It seems that 31… Rgf7 was a mistake by Black. The complete game is at the URL: https://www.chess.com/analysis/game/pgn/3CZZ7wUNKQ?tab=analysis

By MS we mean 2, 4, 8, 12, 16, 20, 28, 32, 36, 44, 52, 64, 68, 76, 84, 96, 108, 128, 132, 140,… which gets its name from the fact that the indices of the powers of two are: 1,2,3,5,8,12,… which, upon adding 1, gives 2, 3, 4, 6, 9, 13, … a sequence… Continue reading Exploration of the Minecraft Sequence

Subject: Analysis and Computation of a Unique Number Sequence Date: June 10, 2023 This memo is to document the findings from the analysis of a specific number sequence. The sequence begins with 2, 4, and 8 and then employs a unique iterative process for generating subsequent numbers. This generation process includes identifying the last number,… Continue reading Memo by ChatGPT-4

What follows is based on the Pari/gp code by ChatGPT-4 at the web page entitled “Pari/gp code to compute unusual sequence from a Turing machine” here:https://pastebin.com/cJSaZy5J The sequence in question is from a simulation of a Turing machine, and by simulating the TM, I computed several thousand termsof this presumptive sequence, which can be found… Continue reading Deciphering Complexity: An Exploration of Turing Machine Sequences

A software stack refers to a collection of software subsystems or components needed to create a complete platform such that no additional software is needed to support applications. Applications are said to “run on” or “run in” the resulting platform. The concept can be visualized as a stack of layers, with each layer being a… Continue reading ChatGPT-4 explains software stacks