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A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

%I A106283 #22 Feb 16 2025 08:32:57
%S A106283 2,5,11,13,31,43,53,79,83,89,97,103,109,131,139,151,197,199,229,233,
%T A106283 239,251,257,271,283,313,317,347,359,367,379,389,433,443,461,479,487,
%U A106283 521,569,571,577,593,599,601,617,631,641,643,647,659,673,677,719,769,797
%N A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.
%C A106283 This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.
%H A106283 Robert Israel, <a href="/A106283/b106283.txt">Table of n, a(n) for n = 1..10000</a>
%H A106283 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%p A106283 Res:= NULL: count:= 0: p:= 0:
%p A106283 P:= x^4 - x^3 - x^2 - x - 1:
%p A106283 while count < 100 do
%p A106283   p:= nextprime(p);
%p A106283   if [msolve(P,p)] = [] then
%p A106283     Res:= Res, p; count:= count+1;
%p A106283   fi
%p A106283 od:
%p A106283 Res; # _Robert Israel_, Mar 13 2024
%t A106283 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, 200}];Prime[Flatten[Position[t, 0]]]
%o A106283 (Python)
%o A106283 from itertools import islice
%o A106283 from sympy import Poly, nextprime
%o A106283 from sympy.abc import x
%o A106283 def A106283_gen(): # generator of terms
%o A106283     p = 2
%o A106283     while True:
%o A106283         if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
%o A106283             yield p
%o A106283         p = nextprime(p)
%o A106283 A106283_list = list(islice(A106283_gen(),20)) # _Chai Wah Wu_, Mar 14 2024
%Y A106283 Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of Lucas 4-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible in mod p).
%K A106283 nonn
%O A106283 1,1
%A A106283 _T. D. Noe_, May 02 2005
%E A106283 Name corrected by _Robert Israel_, Mar 13 2024