S000476


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

S000476

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. A000078A073817A106280A106283, S000475.

nonn

T. D. Noe, Feb 11 2015

© Tony D Noe 2014-2015