A primality test for Kp^n + 1 numbers and a generalization of Safe primes and Sophie Germain primes

In this paper, we provide a generalization of Proth’s theorem for integers the form Kp^n+1. particular, primality test that requires modular exponentiation (with proper base a) similar to Fermat’s without computation any GCD’s. We also two tests increase chances proving Kp^n+1 primes. As corollaries, three families N whose can be certified only by a^{N-1} \equiv 1 \pmod (Fermat’s test). One these is identical Safe primes (since N-1 has large prime factor same as primes). Therefore, considered them and defined a-Safe address some questions regarding distribution those numbers conjecture about their generative a-Sophie Germain which seems true even if are dealing with 100, 1000, or 10000 digits