Conjectures in number theory

Abstract

Prime numbers, whose properties are important subjects in mathematics, are also fundamental in computer science notably in IT security, Cryptocurrencies as Bitcoin and Blockchain, cryptography, Code theory notably Error detection codes, integer factorization, and random number generation. Finding prime numbers is too an active area of research in mathematics. There are many methods for identifying and generating them and many primality tests which are often complex and expensive in terms of calculation time, and many conjectures and theorems related to prime numbers, such as the prime number theorem, Goldbach's conjecture, and the Riemann hypothesis. My objective in this work is to propose two conjectures : $1)$ the integer $1+3.2^{20n}$ is not a prime number for all $n=0$ or $1$ mod $3$, and $2)$ the integer $1+3.2^{4+20n}$ is not a prime number for all $n=2$ mod $3$. Keywords : Prime numbers; IT security; Cryptography.