Found with PFGW Version 3.6.0

4977^9008-9008^4977 is 2-PRP! (51.2988s+0.0043s)
4977^9008-9008^4977 is 3-PRP! (50.8428s+0.0046s)
4977^9008-9008^4977 is 5-PRP! (51.0809s+0.0042s)
4977^9008-9008^4977 is 7-PRP! (50.7709s+0.0046s)
4977^9008-9008^4977 is 11-PRP! (51.0190s+0.0060s)

Primality testing 4977^9008-9008^4977 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Running N-1 test using base 17
Running N+1 test using discriminant 23, base 2+sqrt(23)
Calling N-1 BLS with factored part 0.06% and helper 0.00% (0.18% proof)
4977^9008-9008^4977 is Fermat and Lucas PRP! (302.8879s+0.0044s)

To be completed...