Found with PFGW Version 3.7.5

3878^5825-5825^3878 is 2-PRP! (15.0905s+0.0024s)
3878^5825-5825^3878 is 3-PRP! (15.0161s+0.0024s)
3878^5825-5825^3878 is 5-PRP! (15.1783s+0.0026s)
3878^5825-5825^3878 is 7-PRP! (15.0870s+0.0019s)
3878^5825-5825^3878 is 11-PRP! (15.2496s+0.0023s)

Primality testing 3878^5825-5825^3878 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N-1 test using base 17
Running N+1 test using discriminant 23, base 1+sqrt(23)
Calling N+1 BLS with factored part 0.07% and helper 0.00% (0.23% proof)
3878^5825-5825^3878 is Fermat and Lucas PRP! (105.4562s+0.0042s)

To be completed...