This is a PRP value of the Ramanujan Tau function (LR number).
http://dx.doi.org/10.1007/s11139-012-9420-8

Checked with PFGW (v3.7.10) in all odd prime bases < 20

lucasU(-11145268112466978,773^11,34961) is 3-PRP! (8549.8619s+0.0321s)
lucasU(-11145268112466978,773^11,34961) is 5-PRP! (8573.7899s+0.0324s)
lucasU(-11145268112466978,773^11,34961) is 7-PRP! (8566.4379s+0.0320s)
lucasU(-11145268112466978,773^11,34961) is 11-PRP! (8592.3885s+0.0321s)
lucasU(-11145268112466978,773^11,34961) is 13-PRP! (8579.6782s+0.0321s)
lucasU(-11145268112466978,773^11,34961) is 17-PRP! (8595.4986s+0.0321s)
lucasU(-11145268112466978,773^11,34961) is 19-PRP! (8598.5628s+0.0322s)

and Pari/GP (v2.5.5)

? ispseudoprime(abs(lr(773,34961)))
time = 95h, 24min, 29,400 ms.
%21 = 1

To be completed...