This paper introduces a novel approach employing periodic functions for the comprehensive analysis of prime numbers. The method encompasses primality testing, factor counting and listing, prime distribution calculation, and the determination of the Nth prime. The exposition of the technique is presented in a clear and sequential manner, guiding the reader through each step with explicit equations. Graphs are strategically incorporated between crucial stages to facilitate a rapid and intuitive visualization of the rationale and outcomes of each maneuver. The paper concludes with concise reflections and ongoing inquiries into the potential applications and refinements of the proposed method.

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.

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.

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.

The Riemann Hypothesis remains one of the most critical unsolved problems in mathematics, proposing that all non-trivial zeros of the Riemann zeta function are located on the critical line. This paper explores unique fractal structures within the zeta function, drawing comparisons with the Mandelbrot set and the Smith chart, which together illuminate possible connections between prime distribution, electromagnetic symmetry, and gravitational principles.

The integration of Artificial Intelligence (AI) into education has the potential to revolutionize traditional teaching and learning methods. AI can offer personalized learning experiences, streamline administrative tasks, enhance feedback mechanisms, and provide robust data analysis. Numerous studies have demonstrated the positive impact of AI on both student outcomes and teacher efficiency. However, caution must be exercised when implementing AI in education, considering potential risks and ethical dilemmas. It is essential to use AI as a tool to support human educators rather than replace them entirely. The adoption of AI in education holds the promise of creating more inclusive and effective learning environments, catering to students of diverse backgrounds and abilities. As AI technology continues to advance, the education sector can anticipate even more innovative applications, further shaping the future of learning. This abstract provides an overview of the multifaceted landscape of AI in education, highlighting its potential benefits, associated challenges, and the importance of responsible integration.

 This paper introduces a novel robot parallel evolution design algorithm , leveraging the concept of a module network, to optimize the learning process of collision avoidance, approach, and wall switching behaviors in evolutionary robots. The proposed algorithm is validated and tested, demonstrating its efficacy in enabling evolutionary robots to autonomously exhibit behaviors such as collision avoidance, movement, repli-cation, and attack. The learning methodology focuses on refining the neural network-based strategies for collision avoidance, approach, and wall switching behaviors. The evolutionary robots, operating in a simulated environment, show-case the ability to adapt and enhance their performance over time. The simulation environment includes randomly generated rectangular obstacles with varying side lengths, strategically placed to represent real-world challenges. Additionally, the environment features randomly scattered approach targets, serving as goals for the robots. The modular design of the neural network allows for the integration of fundamental behaviors such as collision avoidance and approach, enabling a progressive enhancement of the robot’s capabilities. As the neural network evolves, the robots demonstrate an increasingly sophisticated ability to navigate their surroundings, avoid obstacles, approach targets, and adapt to dynamic scenarios. Through extensive simulations, the proposed algorithm proves effective in training evolutionary robots to navigate complex environments autonomously. The study contributes to the field of evolutionary robotics by presenting a modular neural network approach that enables the gradual acquisition and integration of diverse behaviors, showcasing the potential for autonomous and adaptive robotic systems in dynamic and challenging environments.Â

This abstract delves into the intricate relationship between the abstract concept of intelligence and its embodiment, drawing attention to the lack of coherence in understanding intelligence and its implications for artificial intelligence (AI). Intelligence, inherently grounded in human experience, attempts to transcend its embodiment, posing challenges for the development of artificial general intelligence (AGI). The concept of embodiment extends beyond physical instantiation, encompassing the interdependence of an autopoietic system on its environment. The pursuit of autonomy and general capability in AGI necessitates the recreation of the organism’s natural condition of embodiment. However, the feasibility , controllability, and overall advantages of such artificial embodiment remain uncertain. This abstract explores the complex interplay between intelligence, embodiment, and the quest for AGI, raising critical questions about the path forward in the development of intelligent systems.Â

 In this paper, we delve into the historical and enduring algebraic conundrum known as the Two Couriers Problem, originally posed by the French mathematician Clairaut in 1746. Over the centuries, this problem has persisted, finding its way into numerous textbooks, journals, and mathematical discussions. One of the remarkable aspects of the Two Couriers Problem is its inherent connection to division by zero, a mathematical operation that has intrigued scholars for generations. Division by zero, a concept laden with complexity and ambiguity, has sparked diverse mathematical approaches. Conventional mathematics regards division by zero as an indeterminate or undefined result. However, alternative methodologies have emerged over time. Transmathematics defines division by zero as either nullity or explicitly positive or negative infinity, offering a different perspective. Saitoh simplifies division by zero as zero, challenging traditional conventions, while Barukˇci´c explores the possibility of defining it as either unity or explicitly positive or implicitly negative infinity. Amidst these varied approaches, the central question persists: which method offers the most effective solution to the enigma of division by zero? To answer this question, we propose utilizing the Two Couriers Problem as an objective benchmark. By subjecting these different mathematical approaches to this historical problem, we aim to rigorously evaluate their efficacy and determine which one stands out as the most viable solution. This paper seeks to unravel the complexities of division by zero through a systematic analysis, utilizing the Two Couriers Problem as a guiding light. By doing so, we endeavor to shed new insights on this age-old mathematical puzzle and contribute valuable perspectives to the ongoing discourse surrounding division by zero and its diverse interpretationÂ

 In this study, we explore the existence of an infinite number of primes represented by the quadratic polynomial 4(Mp − 2)2 + 1 . We propose a hypothesis that considers Fermat primes as a criterion for the infinitude of such primes, where Mp represents Mersenne primes. Additionally, we provide an elementary argument supporting the presence of infinitely many primes in the form , as these primes are a subset of primes of the same form x 2 + 1 . Furthermore, we present a basic argument demonstrating the infinity of Mersenne primes. This paper contributes to the understanding of prime numbers and their intriguing relationships with quadratic polynomials and Fermat primesÂ

This paper introduces and explores the notion of intelligence as a â\euro™considered responseâ\euro™ phenomenon, shedding light on its relevance in defining and understanding human-level intelligence and its various associated facets. The concept of intelligence as a considered response offers a valuable framework for gaining clarity on essential elements of cognition and consciousness. This paper delves into its implications on the mind, self/I, awareness, self-awareness, consciousness, sentience, thoughts and feelings, free will, perception, attention, cognition, expectation, prediction, and learning. Furthermore, the paper argues that embodiment plays a pivotal role in the architecture of Artificial General Intelligence (AGI). Embodiment is proposed as an essential component for AGI to attain grounded cognition, develop a sense of self, and facilitate social learning. This achievement is made possible through direct physical experiences and mental processes, all rooted in the concept of â\euro™considered response.â\euro™ By adopting the â\euro™considered responseâ\euro™ framework, this paper provides a comprehensive perspective on intelligence and its multifaceted dimensions. It offers insights that can contribute to the development of more sophisticated AGI systems and deepen our understanding of human intelligence, paving the way for a more profound exploration of the intersection between artificial and human cognition.

In the pursuit of understanding the enigmatic world of prime numbers, a unique formula has been identified, which can be used to generate prime numbers, with the exception of 2 and 3. This formula is expressed as (n = positive integer, p = prime number), where, intriguingly, the output for other positive integers results in irrational numbers. This enigmatic formula not only reveals prime numbers but also unveils a peculiar pattern related to composite numbers derived from prime factors. These composite numbers, which arise as exceptions in the context of prime generation, exhibit regularity and may offer new insights into the interconnected of prime and composite numbers. This discovery promises to broaden our understanding of the intricate world of number theory and provides a fresh perspective on the nature of prime numbers. Further exploration of this formula may uncover deeper mathematical principles and unlock novel avenues in number theory research. p =√1 + 24n