Found with PFGW Version 3.7.5

11008^12697+12697^11008 is 2-PRP! (47.8003s+0.0027s)
11008^12697+12697^11008 is 3-PRP! (48.1464s+0.0026s)
11008^12697+12697^11008 is 5-PRP! (47.8131s+0.0026s)
11008^12697+12697^11008 is 7-PRP! (47.7615s+0.0026s)
11008^12697+12697^11008 is 11-PRP! (48.2421s+0.0026s)

Primality testing 11008^12697+12697^11008 [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 0.02% and helper 0.02% (0.08% proof)
11008^12697+12697^11008 is Fermat and Lucas PRP! (277.9158s+0.0028s)

To be completed...