Quote:
Originally Posted by PawnProver44
For example, I want to find a prime of the form 43*71^n +300, (for large primes)

...there are no primes of the form 43*71^n+300. 7 divides all of this sequence.
Take (mod 7) of 43*71^n+300. To do that you can take mod 7 of each coefficient: 1*1^n+6, which is 7 = 0 (mod 7).
Anyway, this is just a poor example. The bigger problem is that if you find large enough (say, > 50,000 digits) prime candidates (probable primes), you will not be able to prove that they are prime. If you still want to search for these large PRPs, use
srsieve to sieve. You can submit large PRPs to
PRP top.