Other than the fact that prime numbers are infinite?
By rule, you mean one that predicts primes then yes they all fail.
Not just prediction of prime numbers, but any regularity that humans can comprehend. The regularity must be directly operated on primes rather than on another set of numbers, such as the statement that every even number is the sum of two primes, or every even number is the sum of a prime and a semi prime. Also, the regularity can't be probabilistic as in the Prime Number Theorem.
No, it is not true. How could it be?
This page has many examples of formulas that generate all the primes (and only the primes).
Do any of them take prime numbers as inputs?
I don't quite understand your question -- are you asking for something like a formula that takes a number as input, and returns 1 if it is prime and 0 if it is non-prime?
If so, take a look at Wilson's Theorem (or at the first formula on the page I just linked to).
Wilson's formula is very interesting. Thanks for the info.
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