Right... that's what I meant about irrational numbers. I'm not trying to rain on the parade of the hobby of it all, but if there were an actual pattern, primes would not be useful in cryptography. The hobby of trying to find "the pattern" is Sunday morning armchair alchemy. Nothing wrong with it, and maybe there's a pattern to be found one day. Great. Whether there is or isn't is only relevant given that we don't know what that pattern is yet. It doesn't, as some imagine, imply the existence of a higher order intelligence.
0.1234567891011121314151617181920212223...
Yet it has so much structure that you can easily figure out what any digit 'n' will be.
> The hobby of trying to find "the pattern" is
I have no idea what you are talking about with "the pattern".
> if there were an actual pattern, primes would not be useful in cryptography
Why do you think there are no patterns and that all of the following people are wrong?
* "Patterns in prime numbers" - https://mathsreach.org/Patterns_in_prime_numbers
* "Patterns in the Primes" - https://www.maa.org/meetings/calendar-events/patterns-in-the...
* "Math Mornings at Yale: The Patterns in the Primes, with Andrew Granville" - https://www.youtube.com/watch?v=pO7Egc5Dtqs
* "New Pattern Found in Prime Numbers" - https://phys.org/news/2009-05-pattern-prime.html
* '"Patterns in the Primes" by Stephanie Hanson - Week 5 - MathSoc Public Lecture' - https://www.youtube.com/watch?v=VkbPhjU-S9Q
Note that none of these refer to a definite "the pattern."
Primes are useful in the RSA cryptosystem because modulo exponentiation is cheap while in general factoring the product of two large prime numbers is far less tractable - as far as we know.
There are so many patterns in primes that cryptosystems have been broken by choosing the initial primes poorly. See https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Faulty_key_... .
Primes have lots of order and patterns, so references to chaos and "no such order exists" aren't really relevant.
As to "why" - this is recreational mathematics. It's because people find it fun or engaging. And it isn't special - recreational mathematics covers far more than just primes. There's a lot of people who love calculating digits of pi, as one clear example.