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A106281 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros.

%I A106281 #17 Feb 16 2025 08:32:57
%S A106281 691,733,3163,4259,4397,5419,6637,6733,8009,8311,9803,11731,14923,
%T A106281 17291,20627,20873,22777,25111,26339,27947,29339,29389,29527,29917,
%U A106281 34123,34421,34739,34757,36527,36809,38783,40433,40531,41131,42859,43049
%N A106281 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros.
%C A106281 This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048. The periods of the sequences A001591(k) mod p and A074048(k) mod p have length less than p.
%H A106281 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.
%t A106281 t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^5-x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 5000}];Prime[Flatten[Position[t, 5]]]
%o A106281 (Python)
%o A106281 from itertools import islice
%o A106281 from sympy import Poly, nextprime
%o A106281 from sympy.abc import x
%o A106281 def A106281_gen(): # generator of terms
%o A106281     p = 2
%o A106281     while True:
%o A106281         if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==5:
%o A106281             yield p
%o A106281         p = nextprime(p)
%o A106281 A106281_list = list(islice(A106281_gen(),20)) # _Chai Wah Wu_, Mar 14 2024
%Y A106281 Cf. A106278 (number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n)), A106298, A106304 (period of Lucas and Fibonacci 5-step mod prime(n)).
%K A106281 nonn
%O A106281 1,1
%A A106281 _T. D. Noe_, May 02 2005