Finding Pythagorean Primes

It is well known that finding large prime numbers is difficult.i We have no algorithm to find primes although we have algorithms to narrow the search. The two main tools we have are theory and empirical data both supported by computers. But, with increased number size of numbers computation becomes prohibitive in time and resources. For small prime numbers, it is easy as we have exhaustively tested all less than 20 digit numbers through some fast probable prime tests and found no false results.  Above 20 digits, primality proofs are increasingly slower.ii Here, I discuss the efficacy of prime number theory and empirical data. I find the law of prime numbers cannot be expressed as a limit asymptotic to 1. While theory has yielded the closeness between prime numbers and theoretical approximations, the law of prime numbers is best expressed using empirical approximations.