(2^1347781-2^673891+1)/5 is 3-PRP, originally found using LLR ver.3.7.1 for Windows (no factor till 2^51).
This is a Fermat PRP at base 3, 5, 7, 11, 13, 17, 101, 137 - confirmed with PFGW ver.1.2.0 for Windows.
Additionally using the following command with PFGW:
pfgw -l -tc -a1 -q(2^1347781-2^673891+1)/5
I've received the following result:
Primality testing (2^1347781-2^673891+1)/5 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 2+sqrt(7)
Calling N-1 BLS with factored part 0.03% and helper 0.00% (0.08% proof)
(2^1347781-2^673891+1)/5 is Fermat and Lucas PRP! (104628.5231s+0.0022s)

To be completed...