Found with PFGW Version 3.8.0

4009^12938-12938^4009 is 2-PRP! (43.7077s+0.0056s)
4009^12938-12938^4009 is 3-PRP! (43.3668s+0.0059s)
4009^12938-12938^4009 is 5-PRP! (42.1799s+0.0078s)
4009^12938-12938^4009 is 7-PRP! (42.5873s+0.0085s)
4009^12938-12938^4009 is 11-PRP! (42.4947s+0.0054s)

Primality testing 4009^12938-12938^4009 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 17, base 8+sqrt(17)
Calling N-1 BLS with factored part 0.02% and helper 0.01% (0.07% proof)
4009^12938-12938^4009 is Fermat and Lucas PRP! (256.0596s+0.0060s)

To be completed...