Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 2 distinct zeros.
29, 37, 41, 73, 157, 163, 167, 173, 227, 263, 293, 307, 311, 331, 353, 373, 397, 419, 431, 457, 463, 499, 523, 547, 587, 613, 619, 661, 691, 701, 727, 743, 761, 787, 827, 829, 883, 911, 919, 947, 977, 983, 1009, 1051, 1103, 1109, 1123, 1213, 1223, 1237, 1277
1
This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Eric W. Weisstein, MathWorld: Fibonacci n-Step Number
(Mma) s = Table[f = FactorList[x^4 - x^3 - x^2 - x - 1, Modulus -> Prime[n]]; cnt = 0; Do[If[Exponent[f[[i, 1]], x] == 1, cnt++], {i, 2, Length[f]}]; cnt, {n, 250}]; Prime[Flatten[Position[s, 2]]]
Cf. A000078, A073817, A106280, A106283, S000475.
nonn
T. D. Noe, Feb 11 2015