Found with PFGW Version 3.7.5

11835^12584+12584^11835 is 2-PRP! (48.9761s+0.0034s)
11835^12584+12584^11835 is 3-PRP! (48.8130s+0.0039s)
11835^12584+12584^11835 is 5-PRP! (48.8590s+0.0034s)
11835^12584+12584^11835 is 7-PRP! (58.1860s+0.0036s)
11835^12584+12584^11835 is 11-PRP! (54.2973s+0.0031s)

Primality testing 11835^12584+12584^11835 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling N+1 BLS with factored part 0.03% and helper 0.01% (0.11% proof)
11835^12584+12584^11835 is Fermat and Lucas PRP! (278.2846s+0.0033s)

To be completed...