Found with PFGW Version 3.6.0

3434^4999-4999^3434 is 2-PRP! (13.5076s+0.0022s)
3434^4999-4999^3434 is 3-PRP! (13.4375s+0.0022s)
3434^4999-4999^3434 is 5-PRP! (13.4890s+0.0020s)
3434^4999-4999^3434 is 7-PRP! (13.5346s+0.0020s)
3434^4999-4999^3434 is 11-PRP! (13.5054s+0.0022s)

Primality testing 3434^4999-4999^3434 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 37, base 1+sqrt(37)
Calling N+1 BLS with factored part 0.08% and helper 0.05% (0.28% proof)
3434^4999-4999^3434 is Fermat and Lucas PRP! (64.7948s+0.0039s)

To be completed...