Found with PFGW Version 3.7.5

4478^8253-8253^4478 is 2-PRP! (30.9325s+0.0035s)
4478^8253-8253^4478 is 3-PRP! (26.7888s+0.0022s)
4478^8253-8253^4478 is 5-PRP! (26.9599s+0.0029s)
4478^8253-8253^4478 is 7-PRP! (26.8726s+0.0026s)
4478^8253-8253^4478 is 11-PRP! (30.6485s+0.0036s)

Primality testing 4478^8253-8253^4478 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 19
Running N+1 test using discriminant 37, base 1+sqrt(37)
Calling N+1 BLS with factored part 0.05% and helper 0.02% (0.17% proof)
4478^8253-8253^4478 is Fermat and Lucas PRP! (137.5951s+0.0042s)

To be completed...