Found with PFGW Version 3.6.0

782^4977-4977^782 is 2-PRP! (9.8176s+0.0016s)
782^4977-4977^782 is 3-PRP! (9.8155s+0.0018s)
782^4977-4977^782 is 5-PRP! (9.8204s+0.0018s)
782^4977-4977^782 is 7-PRP! (9.8101s+0.0018s)
782^4977-4977^782 is 11-PRP! (9.8116s+0.0019s)

Primality testing 782^4977-4977^782 [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 0.06% and helper 0.04% (0.24% proof)
782^4977-4977^782 is Fermat and Lucas PRP! (46.2445s+0.0038s)

To be completed...