Found with PFGW Version 3.7.5

4000^8301-8301^4000 is 2-PRP! (17.3101s+0.0005s)
4000^8301-8301^4000 is 3-PRP! (16.7102s+0.0005s)
4000^8301-8301^4000 is 5-PRP! (16.6866s+0.0005s)
4000^8301-8301^4000 is 7-PRP! (16.9104s+0.0005s)
4000^8301-8301^4000 is 11-PRP! (16.8064s+0.0005s)

Primality testing 4000^8301-8301^4000 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 13
Running N+1 test using discriminant 31, base 3+sqrt(31)
Calling N+1 BLS with factored part 0.02% and helper 0.00% (0.06% proof)
4000^8301-8301^4000 is Fermat and Lucas PRP! (100.3449s+0.0028s)

To be completed...