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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?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – e-sushi Dec 17 '17 at 13:53
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Sorry, but I think this definition of “n-prime” number is just nonsense.

Mathematicians defined prime numbers the way they did, because it provides a set of numbers with interesting mathematical properties. A simple example is that any positive integer is the product of prime numbers.

With that definition of “n-prime” numbers, 4 would be 3-prime and 8 would not (since it is divisible by 4). Then 8 is not a product of 3-prime numbers.

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.

By the way, I hate that definition of prime number that you quoted, because it gives the feeling that there’s something special with 2, while there is not.

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