A106280 - OEIS

%I #8 Mar 24 2024 14:55:25

%S 137,179,653,859,991,1279,1601,1609,2089,2437,2591,2693,2789,2897,

%T 3701,3823,3847,4451,4691,4751,4919,5431,5479,5807,5903,5953,6203,

%U 6421,6781,6917,7253,7867,8317,9187,9277,9533,9629,9767,9907,9967,10009,10079

%N Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 4 distinct zeros.

%C This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. The periods of the sequences A000078(k) mod p and A073817(k) mod p have length less than p.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>

%t t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 1600}];Prime[Flatten[Position[t, 4]]]

%Y Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of 4-step sequence mod prime(n)).

%K nonn

%O 1,1

%A _T. D. Noe_, May 02 2005