Prime numbers play an important role in information security and especially in the encryption of messages (see cryptography). They are often used in asymmetric cryptosystems such as public-key encryption schemes. Important examples are the Diffie-Hellman key exchange, the RSA cryptosystem used in OpenPGP, among others, the Elgamal cryptosystem and the Rabin cryptosystem. In these, keys are calculated from large, randomly generated prime numbers that must remain secret.
Such algorithms are based on one-way functions that can be executed quickly, but whose inversion is practically impossible to compute with currently known technology. However, new information technologies, for example quantum computers, could change that. The unsolved P-NP problem is related to this. The P-NP problem (also P≟NP, P versus NP) is an unsolved problem in mathematics, specifically complexity theory in theoretical computer science. The question here is whether the set of all problems that can be solved quickly (P {\displaystyle P} P) and the set of all problems for which one can quickly check a proposed solution for correctness (N P {\displaystyle NP} NP) are identical.
Some species of animals and plants (e.g., certain cicadas or spruce trees) reproduce especially strongly in cycles of primes (say, every 11, 13, or 17 years) to make it difficult for predators to adjust to the mass occurrence.
What I am concerned with in this topic is the answering of the question and the following straight argumentation, whether and why the 1 should be counted again to the prime numbers or not. So basically I am concerned with logic and with straight arguments. It is - as I already said - similar to the question whether 1 and 0.999… are equal or not (I say: they are not).
The more ridiculous than serious argument that “1 is not allowed to be a prime number, because 1 is divisible only by itself and 1” actually means that 1 is a prime number, because the definition that “a prime number is divisible only by itself and 1” also applies to 1.
But then it is said that every prime number has two different divisors, but the number 1 has only itself. Yes, but the original definition does not say anything about the separability of the divisors of a prime number, but only that a prime number is divisible only by itself and by 1.
The original definition has been changed so that one can claim afterwards that another definition than the valid one is “too complicated”. In reality, it’s the other way around.