Authors: Sofiia Ivanko, & Peter Farag*
Published Date: 25 March, 2025
Cite: Ivanko and P. Farag, “Proof of the Riemann Hypothesis and the Modified Collatz Conjecture Using the Sophy-Peter Mathematical Framework,” Crystal Journal of Artificial Intelligence and Applications, vol. 1, no. 1, p. 01-33, 2025.
Abstract
In this article, we present solutions for one of the oldest mathematical problems, the Collatz Conjecture, and provide a proof for the Riemann Hypothesis, utilizing a new number theory based on newly discovered number properties, which will also be presented in this paper. This became possible through the Sophy-Peter mathematical framework, built upon this new number theory. The Collatz Conjecture will be disproven, but as an alternative, the Oscillating Theorem will be introduced, with its correctness proven within this article. Furthermore, we present the general version of the Riemann zeta function, developed based on the new number theory. The correctness of this function is verified by comparing it with existing results of the zeta function. Using this approach and the Sophy-Peter framework, we have successfully proven the Riemann Hypothesis, long considered a millennium problem. Moreover, since the general zeta function is proven, this implies the correctness of the new number theory and the Sophy-Peter framework.
Keywords
Sophy-Peter Framework, New Number Theory, Modified Collatz Conjecture, Oscillating Theorem, General Zeta Function, Proof of the Riemann Hypothesis
Authors: Siddarth Laxminarayanan*
Published Date: 05 June, 2025
Cite: S. Laxminarayanan, “The Convergence of Quantum Computing and AI in Defence: Shaping the Future of Global Security,” Crystal Journal of Artificial Intelligence and Applications, vol. 1, no. 1, pp. 01–06, 2025.
Abstract
Quantum Computing (QC) and Artificial Intelligence (AI) are emerging technologies poised to significantly transform global defense strategies. With exceptional computational capabili-ties, Quantum Computing has profound implications for military cryptography and secure communications, potentially altering national security paradigms. Simultaneously, Artificial Intelligence contributes advanced predictive analytics, strategic modeling, and autonomous system functionalities, enhancing real-time intelligence and operational effectiveness in mili-tary contexts. This convergence between QC and AI offers unprecedented opportunities for strengthening defense systems; however, it also introduces critical challenges related to cyber-security vulnerabilities, ethical considerations in autonomous military operations, and complex geopolitical tensions. Utilizing scenario planning, this article examines three plausible future scenarios: the High-Integration scenario, depicting rapid and comprehensive incorporation of QC and AI into military systems; the Regulated-Adoption scenario, emphasizing international regulations and collaborative oversight mechanisms; and the Ethical Constraints scenario, highlighting stringent ethical guidelines limiting certain military applications. Each scenario illustrates distinct shifts in military capabilities, cybersecurity posture, and the dynamics of autonomous system deployment, while emphasizing the necessity for robust governance frameworks. Moreover, the study underscores the essential role of international collaboration and comprehensive ethical guidelines in responsibly navigating the potential benefits and in-herent risks associated with these advanced technologies. Ultimately, managing the integra-tion of Quantum Computing and Artificial Intelligence within defense sectors effectively requires proactive global cooperation, strategic regulatory frameworks, and ethical oversight to safeguard international security and stability.
Keywords
Quantum Computing, Artificial Intelligence, Governance, Ethics, Defence Technology, Mili-tary Strategy, Global Security, Cryptography, Autonomous Systems
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