Algorithms for Certain Computational Mathematics Problems

This dissertation presents algorithms and explores diverse aspects of number theory, cryptographic systems, and partition theory. In Chapter Two, attention is focused on enhancing the security of the extended Rabin cryptosystem by incorporating multiple prime numbers into the encryption process, thereby increasing the complexity of decryption and fortifying resilience against quantum computing threats. Additionally, experimental results corroborate the efficacy of proposed algorithms, aligning closely with existing decryption methods while offering enhanced versatility. Chapter Three presents a detailed exploration of sums of powers of arithmetic progressions, offering simplified formulas and algorithms for efficient computation, leveraging Stirling and Eulerian numbers. A comparison with existing methods underscores the computational efficiency of the proposed approaches. In Chapter Four, properties and algorithms related to Ramanujan-type cubic equations are elucidated, showcasing a comprehensive computational methodology and its application through examples and cubic Shevelev sums. Chapter Five extends the understanding of Leonardo sequences and second-order non-homogeneous recursive sequences, unveiling novel identities and combinatorial results. These findings are applied to investigate series representations, enriching the discourse on number theory. Lastly, Chapter Six investigates the representation of positive odd integers as the sum of arithmetic progressions, building upon historical and contemporary works to provide theorems and efficient algorithms for computing such representations. This dissertation contributes to diverse areas within mathematics, cryptography, and computational methods, promising new avenues for exploration and application.