Found with PFGW Version 3.7.5

7800^12253+12253^7800 is 2-PRP! (97.1721s+0.0044s)
7800^12253+12253^7800 is 3-PRP! (95.3856s+0.0047s)
7800^12253+12253^7800 is 5-PRP! (95.7976s+0.0045s)
7800^12253+12253^7800 is 7-PRP! (93.9331s+0.0045s)
7800^12253+12253^7800 is 11-PRP! (93.5951s+0.0048s)

Primality testing 7800^12253+12253^7800 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 17, base 8+sqrt(17)
Calling N-1 BLS with factored part 0.02% and helper 0.00% (0.05% proof)
7800^12253+12253^7800 is Fermat and Lucas PRP! (476.6037s+0.0046s)

To be completed...