Found with PFGW Version 3.7.5

3444^8299-8299^3444 is 2-PRP! (16.7497s+0.0023s)
3444^8299-8299^3444 is 3-PRP! (16.6107s+0.0015s)
3444^8299-8299^3444 is 5-PRP! (16.7469s+0.0016s)
3444^8299-8299^3444 is 7-PRP! (16.4294s+0.0014s)
3444^8299-8299^3444 is 11-PRP! (16.5611s+0.0014s)

Primality testing 3444^8299-8299^3444 [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 0.05% and helper 0.03% (0.18% proof)
3444^8299-8299^3444 is Fermat and Lucas PRP! (80.4955s+0.0029s)

To be completed...