Found with PFGW Version 3.7.5

3414^7519-7519^3414 is 2-PRP! (30.6217s+0.0025s)
3414^7519-7519^3414 is 3-PRP! (30.5421s+0.0036s)
3414^7519-7519^3414 is 5-PRP! (30.9796s+0.0027s)
3414^7519-7519^3414 is 7-PRP! (30.2409s+0.0026s)
3414^7519-7519^3414 is 11-PRP! (30.0676s+0.0027s)

Primality testing 3414^7519-7519^3414 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N+1 test using discriminant 11, base 3+sqrt(11)
Calling N-1 BLS with factored part 0.03% and helper 0.02% (0.11% proof)
3414^7519-7519^3414 is Fermat and Lucas PRP! (152.0777s+0.0048s)

To be completed...