%I #21 Mar 24 2024 14:55:15

%S 2,5,11,13,31,43,53,79,83,89,97,103,109,131,139,151,197,199,229,233,

%T 239,251,257,271,283,313,317,347,359,367,379,389,433,443,461,479,487,

%U 521,569,571,577,593,599,601,617,631,641,643,647,659,673,677,719,769,797

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

%C This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.

%H Robert Israel, <a href="/A106283/b106283.txt">Table of n, a(n) for n = 1..10000</a>

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

%p Res:= NULL: count:= 0: p:= 0:

%p P:= x^4 - x^3 - x^2 - x - 1:

%p while count < 100 do

%p p:= nextprime(p);

%p if [msolve(P,p)] = [] then

%p Res:= Res, p; count:= count+1;

%p fi

%p od:

%p Res; # _Robert Israel_, Mar 13 2024

%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, 200}];Prime[Flatten[Position[t, 0]]]

%o (Python)

%o from itertools import islice

%o from sympy import Poly, nextprime

%o from sympy.abc import x

%o def A106283_gen(): # generator of terms

%o p = 2

%o while True:

%o if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0:

%o yield p

%o p = nextprime(p)

%o A106283_list = list(islice(A106283_gen(),20)) # _Chai Wah Wu_, Mar 14 2024

%Y 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 nonn

%O 1,1

%A _T. D. Noe_, May 02 2005

%E Name corrected by _Robert Israel_, Mar 13 2024