Month: January 2024

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)