Found with PFGW Version 3.7.5

5324^8047-8047^5324 is 2-PRP! (32.7931s+0.0025s)
5324^8047-8047^5324 is 3-PRP! (34.7781s+0.0035s)
5324^8047-8047^5324 is 5-PRP! (28.2028s+0.0026s)
5324^8047-8047^5324 is 7-PRP! (37.9115s+0.0032s)
5324^8047-8047^5324 is 11-PRP! (25.8294s+0.0024s)

Primality testing 5324^8047-8047^5324 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 6+sqrt(7)
Calling N-1 BLS with factored part 0.03% and helper 0.02% (0.11% proof)
5324^8047-8047^5324 is Fermat and Lucas PRP! (194.5417s+0.0057s)

To be completed...