On the infinity of Germain’s extended prime numbers: a novel approach: Sobre la infinitud de los primos extendidos de Germain: un nuevo enfoque - Miramontes de León, Gerardo (2023)
The conjecture about the infinity of Germain primes, that is, those that “if p is prime, 2p+1 is also prime“, is treated in this work following a novel approach. We first observe that there is an infinite number of primes p that are not Germain primes. Therefore, if the number of Germain primes is infinite, there is no bijection with all primes. However, in this work it is shown that by making an extension to Germain’s definition, this bijection is obtained. To achieve this, the definition of ”2p+1” is extended to “kp + (k - 1), with k ≥ 2”, which will be defined as extended Germain primes. This allows us to pose, among others, the conjecture that there is an infinite number of extended Germain primes and their bijection to the infinite set of prime numbers. The last conjecture states that, in the form kp + (k - 1), no prime p falls outside the category of being a Germain prime.