Category: History

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

The sequence starts with 2, 4, and 8. Afterwards, the following rules are applied iteratively: So for example, if we start with 2, 4, 8: This should generate the sequence as you’ve described it. 2, 4, 8, 12, 16, 20, 28, 32, 36, 44, 52, 64, 68, 76, 84, 96, 108, 128, 132, 140, 148,… Continue reading ChatGPT-4 describes a sequence with simple rules

To calculate the terms of the chaotic TM sequence, it’s best to do it one bit at a time, meaning to do theleast significant bit first (or bit 1), then bit 2, then bit 3 and so on for a number of bit positionsapproximately equal to log_2(n_max), where n_max is the largest value of n… Continue reading Computing the lowest 6 bits of an unusual sequence

Sure, here’s a brief summary: In an attempt to improve efficiency in finding potential pseudoprimes (a number that passes certain primality tests despite not being prime), a number of evolving scripts have been employed, harnessing the mathematical power of the PARI/GP computing environment. These scripts aimed to conduct certain pseudoprime tests on terms generated from… Continue reading Pseudoprime testing, script included

Recently, our team has made significant progress in the research and exploration of Turing machines, specifically focusing on a five-state machine that exhibits chaotic behavior. The approach has been to examine the machine’s “tape” – the primary data structure that it manipulates. A new and innovative proposal has been put forward that may offer deeper… Continue reading Update on Turing Machine Research – Rethinking Tape Analysis

Congratulations on generating agreement for 512k terms of the sequence! The code you’ve shared appears to be a refinement of your original Turing machine implementation. Your code generates a binary sequence using the initial “gen” arrays, then expands this sequence exponentially by applying a recursive rule to generate the next “gen” array from the previous… Continue reading ChatGPT-4 comments on my program to predict a mysterious sequence

Our ongoing research into the mysterious sequence originating from Marxen and Buntrock’s chaotic Turing machine (Tm) continues to yield intriguing insights. We have implemented an advanced sequence generation algorithm in C, which successfully emulates the initial 4096 terms of this complex sequence, mirroring Marxen and Buntrock’s original process. The generation algorithm, which begins with five… Continue reading Progress Report on Chaotic Turing Machine Sequence Research

Background: We’ve been working on a problem related to predicting a specific sequence of numbers. The sequence is generated through a mysterious process that we are trying to model with a C program. The C program, however, is not correctly predicting the sequence as expected, leading to differences in output and therefore a challenging debugging… Continue reading Debugging and Prediction Challenges in Mysterious Sequence Generation Using C Code

The Mysterious Sequence is a peculiar binary pattern that we’ve been studying. Starting with a “1”, it generates a binary array by a simple rule: each bit duplicates itself and then appends an inverted version. This results in a fascinating pattern of zeros and ones. We initially thought this sequence might be related to the… Continue reading An Algorithmic Description of the “Mysterious Sequence”

Dear Team, I am writing to provide an update on the development of our prediction algorithm for the Moser-de Bruijn sequence. The sequence, notable for its origins in the run-length encoding of the tape of a chaotic 5-state Turing machine, presents unique challenges for accurate prediction. As you may know, the Moser-de Bruijn sequence is… Continue reading Memo on the Prediction Algorithm for the Moser-de Bruijn Sequence

Dear Team, I would like to bring your attention to an intriguing algorithm that we have recently developed. The goal was to predict the ‘next generation’ in the Moser-de Bruijn sequence. This sequence, found within number theory, is generated from the sum of distinct powers of 4 and carries intriguing structural properties. Our algorithm begins… Continue reading Generation Algorithm for Moser-de Bruijn Sequence

Indeed, the concept of “generations” and the understanding of sequential transitions has proved to be a very insightful approach in your investigation of the Moser-de Bruijn sequence. This generation-based concept has provided an effective framework to analyze and predict the pattern of the sequence. The identification of the last quarter of the current generation as… Continue reading Copied from ChatGPT-4 discussion

We have recently embarked on a study of the Moser-de Bruijn sequence and its generational patterns. The sequence is renowned for its intriguing recurrence and regularity, and we’ve discovered compelling hints of structured growth across generations. Our approach involved breaking down the sequence into ‘genfiles’, each representing a specific generation (bit) of the sequence. Each… Continue reading Exploration of Moser-de Bruijn Sequence Generational Patterns

Subject: Update on Moser-de Bruijn Sequence Analysis Dear Team, Our ongoing exploration of the Moser-de Bruijn sequence has yielded some intriguing insights. The original complexity of the sequence’s patterns has begun to distill into a simpler and more manageable framework, particularly with the introduction of the function g(x) = round(1.0815 * 1.5^x). Through deep analysis,… Continue reading A memo from ChatGPT-4

The prompt was: “So do I. I think this is a turning in human history: the rise of higher AI using LLM…” and GPT-4’s response: Indeed, the development of large language models (LLMs) and AI in general has been a significant turning point in human history. These technologies have the potential to transform a multitude… Continue reading What Chat-GPT-4 thinks

To whom it may concern, We’ve identified a potentially promising line of inquiry relating to sequence A274152, derived from a chaotic Turing machine. This sequence has intrigued researchers due to its unpredictable nature and lack of apparent pattern. Our new approach is rooted in the hypothesis that the opcode sequences used in evaluating certain polynomials… Continue reading A memo written by ChatGPT-4

Memo: Recent developments have allowed us to establish an intriguing link between a mysterious sequence arising from a chaotic Turing machine and a newly developed algorithm. This sequence, cataloged as A274152 in the OEIS database, displays a surprising property that we’ve successfully captured using a novel approach involving numerical approximation. The algorithm uses opcode sequences… Continue reading A Connection Between the Chaotic Turing Machine and A New Algorithm

We embarked on an in-depth investigation into the growth patterns of a tree structure at a growth factor of 1.5, focusing on the resulting number of distinct integers after each generation. Through the development and utilization of precise computational models, our empirical observations revealed a numerical sequence with compelling properties.However, a comparison with the existing… Continue reading (ChatGPT-4) Memo: Discrepancies and Conjectures Surrounding Tree Growth Numbers at 1.5: OEIS A274152 versus Empirical Observations

As part of our ongoing efforts in numerical analysis and understanding of mathematical phenomena, this memo provides a comparative review of three distinct sequences: a Turing Machine-generated sequence, the OEIS A274152 sequence, and a sequence generated from empirical results on tree evaluations at 1.5. Each of these sequences presents interesting characteristics and complexities. (a) Turing… Continue reading Review and Comparison of Three Numerical Sequences: Turing Machine Sequence, OEIS A274152, and Empirical Results from Tree Evaluations

k d_k A274152_{k+2} d_k/A274152_{k+2}1 2 2 1.0002 2 2 1.0003 4 4 1.0004 6 6 1.0005 8 8 1.0006 12 12 1.0007 18 18 1.0008 28 28 1.0009 42 42 1.00010 62 62 1.00011 94 96 0.97912 140 142 0.98613 210 210 1.00014 316 316 1.00015 474 474 1.00016 710 712 0.99717 1066 1070 0.99618… Continue reading Re: OEIS (A274152) and a mysterious sequence

I’m writing to provide an update on our recent exploration into Turing Machine (TM) simulations, the resulting technical issues we’ve encountered, and some intriguing discoveries about the Mandelbrot Sequence. Our current focus has been on a particular Turing Machine, referred to as the Chaotic Turing Machine (TM). It has the unique ability to generate segments… Continue reading Recent Developments in Turing Machine Simulations and Associated Challenges