Found with PFGW Version 3.7.5

3677^12222+12222^3677 is 2-PRP! (137.7204s+0.0042s)
3677^12222+12222^3677 is 3-PRP! (135.0369s+0.0063s)
3677^12222+12222^3677 is 5-PRP! (134.3038s+0.0046s)
3677^12222+12222^3677 is 7-PRP! (136.2751s+0.0058s)
3677^12222+12222^3677 is 11-PRP! (135.7333s+0.0078s)

Primality testing 3677^12222+12222^3677 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Running N-1 test using base 13
Running N+1 test using discriminant 19, base 6+sqrt(19)
Calling N-1 BLS with factored part 0.03% and helper 0.00% (0.10% proof)
3677^12222+12222^3677 is Fermat and Lucas PRP! (820.3933s+0.0046s)

To be completed...