Found with PFGW Version 3.8.0

5779^12648-12648^5779 is 2-PRP! (42.4352s+0.0070s)
5779^12648-12648^5779 is 3-PRP! (42.2201s+0.0086s)
5779^12648-12648^5779 is 5-PRP! (42.3105s+0.0082s)
5779^12648-12648^5779 is 7-PRP! (42.2287s+0.0085s)
5779^12648-12648^5779 is 11-PRP! (42.3832s+0.0091s)

Primality testing 5779^12648-12648^5779 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Running N-1 test using base 29
Running N-1 test using base 41
Running N+1 test using discriminant 47, base 2+sqrt(47)
Calling N-1 BLS with factored part 0.05% and helper 0.01% (0.15% proof)
5779^12648-12648^5779 is Fermat and Lucas PRP! (334.6277s+0.0080s)

To be completed...