Fibonacci number
PFGW Fermat Prp b=2,3,5,11 and BLS test :

PFGW Version 20040817.Win_Dev (Beta \\caveat utilitor\\) [FFT v23.8]
Primality testing F(593689) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 13, base 6+sqrt(13)

F(593689) is Fermat and Lucas PRP! (51918.0100s+0.2055s)

Note : Lucas number L(593689) is composite and has factors: 24934939

To be completed...