Found with PFGW Version 3.6.0

4164^6239-6239^4164 is 2-PRP! (22.0689s+0.0040s)
4164^6239-6239^4164 is 3-PRP! (22.0417s+0.0036s)
4164^6239-6239^4164 is 5-PRP! (22.0460s+0.0040s)
4164^6239-6239^4164 is 7-PRP! (22.0049s+0.0036s)
4164^6239-6239^4164 is 11-PRP! (22.0790s+0.0037s)

Primality testing 4164^6239-6239^4164 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 19, base 1+sqrt(19)
Calling N+1 BLS with factored part 0.03% and helper 0.00% (0.10% proof)
4164^6239-6239^4164 is Fermat and Lucas PRP! (103.8899s+0.0037s)

To be completed...