Found with PFGW Version 3.7.5

1010^5939-5939^1010 is 2-PRP! (9.3382s+0.0016s)
1010^5939-5939^1010 is 3-PRP! (11.4251s+0.0017s)
1010^5939-5939^1010 is 5-PRP! (10.9319s+0.0016s)
1010^5939-5939^1010 is 7-PRP! (9.4506s+0.0016s)
1010^5939-5939^1010 is 11-PRP! (12.7193s+0.0016s)

Primality testing 1010^5939-5939^1010 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 7
Running N+1 test using discriminant 19, base 1+sqrt(19)
Calling N+1 BLS with factored part 0.04% and helper 0.01% (0.12% proof)
1010^5939-5939^1010 is Fermat and Lucas PRP! (76.7354s+0.0040s)

To be completed...