Found with PFGW Version 3.6.0

5864^6757-6757^5864 is 2-PRP! (25.0245s+0.0047s)
5864^6757-6757^5864 is 3-PRP! (24.8947s+0.0045s)
5864^6757-6757^5864 is 5-PRP! (25.2031s+0.0028s)
5864^6757-6757^5864 is 7-PRP! (24.9327s+0.0026s)
5864^6757-6757^5864 is 11-PRP! (24.9214s+0.0034s)

Primality testing 5864^6757-6757^5864 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 9+sqrt(7)
Calling N-1 BLS with factored part 0.03% and helper 0.03% (0.14% proof)
5864^6757-6757^5864 is Fermat and Lucas PRP! (122.3381s+0.0033s)

To be completed...