Found with PFGW Version 3.7.5

1308^5305-5305^1308 is 2-PRP! (9.6963s+0.0035s)
1308^5305-5305^1308 is 3-PRP! (8.9515s+0.0024s)
1308^5305-5305^1308 is 5-PRP! (10.7005s+0.0030s)
1308^5305-5305^1308 is 7-PRP! (11.0706s+0.0019s)
1308^5305-5305^1308 is 11-PRP! (10.6762s+0.0019s)

Primality testing 1308^5305-5305^1308 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 2+sqrt(11)
Calling N+1 BLS with factored part 0.04% and helper 0.00% (0.13% proof)
1308^5305-5305^1308 is Fermat and Lucas PRP! (55.1730s+0.0033s)

To be completed...