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A106284 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros.

%I A106284 #21 Feb 16 2025 08:32:57
%S A106284 3,5,7,11,13,17,31,37,41,53,71,79,83,107,151,157,199,229,233,239,241,
%T A106284 257,263,277,281,311,317,331,337,379,389,409,431,433,463,467,521,523,
%U A106284 541,547,557,563,571,577,607,631,659,677,727,769,787,809,827,839,853
%N A106284 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros.
%C A106284 This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048.
%H A106284 Robert Israel, <a href="/A106284/b106284.txt">Table of n, a(n) for n = 1..10000</a>
%H A106284 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%p A106284 P:= x^5-x^4-x^3-x^2-x-1:
%p A106284 select(p -> [msolve(P,p)] = [], [seq(ithprime(i),i=1..10000)]); # _Robert Israel_, Mar 13 2024
%t A106284 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, 200}];Prime[Flatten[Position[t, 0]]]
%o A106284 (Python)
%o A106284 from itertools import islice
%o A106284 from sympy import Poly, nextprime
%o A106284 from sympy.abc import x
%o A106284 def A106284_gen(): # generator of terms
%o A106284     from sympy.abc import x
%o A106284     p = 2
%o A106284     while True:
%o A106284         if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
%o A106284             yield p
%o A106284         p = nextprime(p)
%o A106284 A106284_list = list(islice(A106284_gen(),20)) # _Chai Wah Wu_, Mar 14 2024
%Y A106284 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 sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).
%K A106284 nonn
%O A106284 1,1
%A A106284 _T. D. Noe_, May 02 2005
%E A106284 Name corrected by _Robert Israel_, Mar 13 2024