PFGW:
(41*10^54051-23)/9 is 3-PRP! (184.7187s+0.0018s)
Primality testing (41*10^54051-23)/9 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling N-1 BLS with factored part 0.02% and helper 0.00% (0.07% proof)
(41*10^54051-23)/9 is Fermat and Lucas PRP! (893.7187s+0.0351s)

LLR:
(41*10^54051-23)/9 is base 3-Strong Fermat PRP! (54052 decimal digits) Time : 51.408 sec.
(41*10^54051-23)/9 is strong-Fermat, Lucas and Frobenius PRP! (P = 5, Q = 2, D = 17) Time : 189.264 sec.

To be completed...