This paper delves into the possibility of creating a precise mathematical equation to calculate the exact number of prime numbers below a certain threshold. By utilizing tools from function analysis, series of functions, and innovative methods, the goal is to create a comprehensive formula that can accurately determine the quantity of primes within a specified numerical range. The research explores the theoretical underpinnings of prime numbers, using advanced techniques to improve and expand upon current methodologies. Through a combination of insights from function analysis and series theory, the proposed formula seeks to offer a strong and efficient solution for computing prime counts. This effort not only adds value to the field, but it also enhances our understanding of prime number distribution.

This paper introduces a preliminary concept aimed at achieving Artificial General Intelligence (AGI) by leveraging a novel approach rooted in two key aspects. Firstly, we present the General Intelligent Network (GIN) paradigm, which integrates information entropy principles with a generative network, reminiscent of Generative Adversarial Networks (GANs). Within the GIN network, original multimodal information is encoded as low information entropy hidden state representations (HPPs). These HPPs serve as efficient carriers of contextual information, enabling reverse parsing by contextually relevant generative networks to reconstruct observable information. Secondly, we propose a Generalized Machine Learning Operating System (GML System) to facilitate the seamless integration of the GIN paradigm into the AGI framework. The GML system comprises three fundamental components: an Observable Processor (AOP) responsible for real-time processing of observable information, an HPP Storage System for the efficient retention of low entropy hidden state representations, and a Multimodal Implicit Sensing/Execution Network designed to handle diverse sensory inputs and execute corresponding actions. By combining the GIN paradigm and GML system, our approach aims to create a holistic AGI system capable of encoding, processing, and reconstructing information in a manner akin to human-like intelligence. The synergy of information entropy principles and generative networks, along with the orchestrated functioning of the GML system, presents a promising avenue towards achieving advanced cognitive capabilities in artificial systems. This preliminary concept lays the groundwork for further exploration and refinement in the pursuit of true brain-like intelligence in machines.

The Collatz conjecture, a longstanding mathematical puzzle, posits that, regardless of the starting integer, iteratively applying a specific formula will eventually lead to the value 1. This paper introduces a novel approach to validate the Collatz conjecture by leveraging the binary representation of generated numbers. Each transition in the sequence is predetermined using the Collatz conjecture formula, yet the path of transitions is revealed to be intricate, involving alternating increases and decreases for each initial value. The study delves into the global flow of the sequence, investigating the behavior of the generated numbers as they progress toward the termination value of 1. The analysis utilizes the concept of probability to shed light on the complex dynamics of the Collatz conjecture. By incorporating probabilistic methods, this research aims to unravel the underlying patterns and tendencies that govern the convergence of the sequence. The findings contribute to a deeper understanding of the Collatz conjecture, offering insights into the inherent complexities of its trajectories. This work not only validates the conjecture through binary representation but also provides a probabilistic framework to elucidate the global flow of the sequence, enriching our comprehension of this enduring mathematical mystery.

This paper investigates the Diophantine equation p r + (p + 1) s = z 2 Where p > 3, s ≥ 3 , z is an even integer. The focus of the study is to establish rigorous results concerning the existence of solutions within this specific parameter space. The main result presented in this paper demonstrates the absence of solutions under the stated conditions. The proof employs mathematical techniques to systematically address the case when the prime p exceeds 3, and the exponent s is equal to or greater than 2, while requiring the solution to conform to the constraint of an even z. This work contributes to the understanding of the solvability of the given Diophantine equation and provides valuable insights into the interplay between prime powers and the resulting solutions.

Cosmology and prime number theory relate through the analysis of metric space in non-decelerative universe model with logarithmic integral function. It has been seen in history that the three components also exhibit relationships among each other, particularly through the zeroes of zeta functions, random matrix theory and energy levels in quantum chaotic systems. How large-scale structure of universe is related to some unknown prime number theories via mathematical tools. With this integration, new vistas are opened for refrigerators stored in mathematics and Cosmology.

In this research, we examine a special property of some prime numbers that goes beyond separating primes from odd composites. Rather, this property serves to divide the set of primes into two groups: those having it and those lacking it. Incredibly, these two groups portray an almost equal distribution of primes amongst them. This phenomenon is explored in order to reveal some aspects concerning the basic characteristics and structure of prime numbers. We also provide further insight on what this trait implies within prime number theory through rigorous analysis as well as mathematical inspection which shines a new light on the enigmatic properties of these fundamental mathematical entities.

This paper presents a new proof of the Goldbach conjecture, which is a well-known problem originating from number theory that was proposed by Christian Goldbach back in 1742. Our way gives a simple but deep understanding of the even integers can be written as the sum of two prime numbers Through examining fully we show that every other even integer larger than two will essentially represent itself in form adding up two prime numbers. The revelation of a straightforward and elegant line to this enduring conjecture comes from the use of basic number theory concepts such as; by going a step further and coming up with creative strategies. There is more evidence and sound payments she makes for her assertion as we continue. The centuries-old mathematical puzzle has been solved paving way for the exploration of new possibilities in number theory and we are grateful for the perspective and the persistence accorded us by God, which enabled us to reach this milestone.

The sums of prime components of composite numbers lead to prime numbers, a remarkable phenomena in number theory that is examined in this paper. We present a constructive proof methodology to show that adding the prime components of the resulting composite numbers iteratively converges to prime numbers for any composite number higher than four. By applying logical reasoning and the fundamentals of number theory, we prove that all prime numbers larger than three have a finite number of "prime friends," or composite numbers whose prime factors add up to the prime number itself. We reveal the method by which prime numbers emerge from this iterative process by analyzing partitions of prime numbers and applying the function 2k + 3. This abstract summarizes our investigation into the convergence of prime numbers through the sums of prime factors of composite numbers, demonstrating a significant relationship between primes and composites in number theory. NOTE : We have created the term "prime friend"; it is not a term found in the traditional lexicon of mathematics.

In this article, we discuss the concept of infinity (∞), infinitesimals □ , and a new mathematical operation that expands calculations into the realm of the infinite. This breakthrough allows for the computation of infinite numbers, leading to new possibilities for mathematical research and practical applications. By introducing infinite numbers, numerous advantages arise, such as accurately representing limits, divergences in integrals and infinite series, and extending calculations to infinity. This mathematical advancement opens doors to exploring uncharted territories, offering valuable insights and breakthroughs.

In the pursuit of enhancing human life and safety through artificial intelligence (AI), transparency and predictability emerge as crucial elements. The potential for self-awareness in AI, we argue, hinges on its autonomy in decision-making. To address this, we propose three fundamental laws of AI, emphasizing their practical implementability. These laws aim to establish a framework that ensures transparency, predictability, and independence in decision-making, ultimately contributing to the responsible development and deployment of artificial intelligence for the benefit of humanity.

When discussing the Diophantine quintic equation p(a 5 + b 5) = q(c 5 + d 5) where p is a prime and q is an integer, there is a clear gap between the mathematical literature and online forums. Because of the intrinsic complexity of parameterizing fifth-degree equations, this equation remains largely unexplored. In this work, we approach this quintic problem through algebraic methods with the goal of providing numerical solutions that illuminate its properties and behaviors.Our research uncovers intriguing patterns and connections that shed light on the enigmatic nature of Diophantine quintic equations in this particular form.

Who dominates the destiny of the world, humans or artificial intelligence (AI)? This question strikes at the very heart of contemporary humanity's existential anxieties about its future. If we want to seriously consider whether or not unfriendly AI 'neurons' pose any threat to human civilisation and humanity's continual existence and evolution in the Universe, we need to know as much as possible about the Universe in which we find ourselves, our place in it, and what cognition, consciousness and mentality really are. How might we combine philosophical, cognitive science and technological perspectives, to explore the evolving relationship between humans and AI, in order to engage and address the questions at the core of this human-AI complex, namely the future of civilisation-what will it look like, who can claim to be our successors, towards what goals and ends? The evolution and development of human cognition as well as the emergence of AI can help us define these potential paths of future development. Where do we stand today, in relation to our own history and development and to the possibilities that artificial intelligence can offer us? The essay explores the ethical, social and existential questions that arise from the increasing automation of artificial intelligence and how it relates to the story of humanity, from its origins to its contemporary cultural expression. It also underscores the significance of holistic approaches to apprehending and addressing the risks that come with AI development. Such approaches should combine findings from various fields namely philosophy, morality, psyche and technology so as to manage a complicated set of problems. To sum up, this summary highlights the critical necessity for sophisticated viewpoints that go beyond simple man versus machine divisions. It is rather proposing a situation where humans use AI as an instrument for improving collective happiness and ensuring responsible management over technological advances and the larger life system.