With pari/gp , I tried to prove the primality of the number $$63638089207891963635134908^{1024}+1$$
I applied the Pocklington test and the output was
? n=63638089207891963635134908^1024+1;
? factorint(n-1)
%19 =
[ 2 2048]
[ 7 1024]
[ 39857 1024]
[ 1803629963 1024]
[31616009371 1024]
? isprime(n,1)
%20 =
[ 2 3 1]
[ 7 2 1]
[39857 2 1]
Apparently, only the three smaller prime factors were used in the certification. I wonder why this certification is valid because apparently, $A=(2^2\cdot 7\cdot 39857)^{1024}$ is used which is even smaller than the fourth root of $n$.
Is there a variant of the Pocklington test justifying this certification with only $3$ of the $5$ prime factors ? Or what else do I miss ?