Found with PFGW Version 3.7.5

5755^12682+12682^5755 is 2-PRP! (42.2878s+0.0029s)
5755^12682+12682^5755 is 3-PRP! (42.6791s+0.0028s)
5755^12682+12682^5755 is 5-PRP! (41.8241s+0.0026s)
5755^12682+12682^5755 is 7-PRP! (41.8547s+0.0026s)
5755^12682+12682^5755 is 11-PRP! (42.7157s+0.0026s)

Primality testing 5755^12682+12682^5755 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N-1 test using base 5
Running N+1 test using discriminant 13, base 1+sqrt(13)
Calling N+1 BLS with factored part 0.02% and helper 0.01% (0.08% proof)
5755^12682+12682^5755 is Fermat and Lucas PRP! (253.8136s+0.0029s)

To be completed...