In this paper we generalize the classical Proth’s theorem and the Miller-Rabin test for integers of the form
N
=
K
p
n
+
1
N=Kp^n+1
. For these families, we present variations on the classical Pocklington’s results and, in particular, a primality test whose computational complexity is
O
~
(
log
2
N
)
\widetilde {O}(\log ^2 N)
and, what is more important, that requires only one modular exponentiation modulo
N
N
similar to that of Fermat’s test.