# Are there stenography benefits to a “n-prime”?

As mentioned in this StackOverflow CodeGolf question, prime numbers can be redefined:

One of my favorite definitions of the prime numbers goes as follows:

• 2 is the smallest prime.
• Numbers larger than 2 are prime if they are not divisible by a smaller prime.

However this definition seems arbitrary, why 2? Why not some other number? Well lets try some other numbers will define n-prime such that

• n is the smallest n-prime.
• Numbers larger than n are n-prime if they are not divisible by a smaller n-prime.

based on this notion, and we find a union of "real primes" and this fictional subset:

• Are there any cryptographic algorithms, stenography approaches that can be adapted to use an n-prime?

• If I look at the case of n-prime == 2, I assume that there are computation and size efficiencies vs other values for n-prime. What other benefits does n-prime == 2 offer?

• Similarly how the primes 3 and 65537 provide cryptographic benefits as RSA exponents, are there other worthwhile values for n-prime?

• Comments are not for extended discussion; this conversation has been moved to chat. – e-sushi Dec 17 '17 at 13:53

For cryptography, prime numbers are often used because $\mathbb Z/p\mathbb Z$ is a finite field. But that’s only true for “real” prime numbers, not for “n-prime” numbers that are not prime numbers.