Infinite Patterns of Primality

Prompt

Conjectures in Number TheoryAhmed AsimiDepartment of Mathematics, Faculty of Sciences, University Ibnou Zohr, Agadir, Morocco Abstract Prime numbers are fundamental in mathematics and essential in IT security, cryptography, blockchain, and error detection codes. Identifying primes is computationally expensive, and numerous theorems and conjectures attempt to describe their properties. This study proposes two conjectures regarding the non-primality of specific integer sequences: is not a prime number for all . is not a prime number for all . Introduction Prime numbers are infinite and crucial in number theory, cryptography, and code theory. Their distribution remains an open research question. The study focuses on primes of the form and establishes two conjectures on their non-primality. Prime Number Generation Methods Eratosthenes Sieve: Eliminates multiples of primes. Fermat’s & Miller-Rabin Tests: Probabilistic methods for primality. Elliptic Curves: Used in cryptographic applications. Integer Factoring: Basis of RSA encryption. Deterministic Tests: AKS primality test confirms primality with certainty but is computationally intensive. Applications of Prime Numbers Cryptography: Secure key generation, encryption (RSA, DSA, DSS), and digital signatures. Computer Security: SSL/TLS certificates, Diffie-Hellman protocol. Cryptocurrencies: Blockchain security. Code Theory: Error detection and correction in communications. Random Number Generation: Essential for cryptographic security. Proposed Conjectures and Proofs Proposition 1: If , then is not prime (divisible by 5). Proposition 2: If , then is not prime (divisible by 7). Conjecture 1: is not prime for all . Conjecture 2: is not prime for all . Conclusion Prime numbers remain a significant research area in mathematics and computer science. The proposed conjectures provide insights into the structure of certain prime-related sequences and contribute to the ongoing study of number theory

Image Details