Primality test for numbers of the form (2p)^(2^n)+1

We describe a primality test for number $M=(2p)^{2^n}+1$ with odd prime $p$ and positive integer $n$. And we also give the special primality criteria for all odd primes $p$ not exceeding 19. All these primality tests run in polynomial time in log$_{2}(M)$. A certain special $2p$-th reciprocity law is used to deduce our result.