Found with PFGW Version 3.7.5

1061^5892-5892^1061 is 2-PRP! (9.5638s+0.0017s)
1061^5892-5892^1061 is 3-PRP! (9.6343s+0.0019s)
1061^5892-5892^1061 is 5-PRP! (11.4120s+0.0024s)
1061^5892-5892^1061 is 7-PRP! (9.7617s+0.0016s)
1061^5892-5892^1061 is 11-PRP! (9.8680s+0.0016s)

Primality testing 1061^5892-5892^1061 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N-1 test using base 11
Running N-1 test using base 13
Running N-1 test using base 29
Running N-1 test using base 37
Running N-1 test using base 61
Running N-1 test using base 71
Running N-1 test using base 73
Running N-1 test using base 89
Running N-1 test using base 107
Running N+1 test using discriminant 113, base 1+sqrt(113)
Calling N+1 BLS with factored part 0.05% and helper 0.01% (0.17% proof)
1061^5892-5892^1061 is Fermat and Lucas PRP! (158.9249s+0.0041s)

To be completed...