Found with PFGW Version 3.7.5

409^10638-10638^409 is 2-PRP! (14.6576s+0.0013s)
409^10638-10638^409 is 3-PRP! (14.6843s+0.0012s)
409^10638-10638^409 is 5-PRP! (14.6450s+0.0016s)
409^10638-10638^409 is 7-PRP! (16.7194s+0.0019s)
409^10638-10638^409 is 11-PRP! (17.0589s+0.0015s)

Primality testing 409^10638-10638^409 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 1+sqrt(11)
Calling N+1 BLS with factored part 0.04% and helper 0.01% (0.12% proof)
409^10638-10638^409 is Fermat and Lucas PRP! (72.9780s+0.0020s)

To be completed...