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%I #16 Mar 24 2024 07:55:47
%S 691,733,3163,4259,4397,5419,6637,6733,8009,8311,9803,11731,14923,
%T 17291,20627,20873,22777,25111,26339,27947,29339,29389,29527,29917,
%U 34123,34421,34739,34757,36527,36809,38783,40433,40531,41131,42859,43049
%N Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros.
%C 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 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^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 (Python)
%o from itertools import islice
%o from sympy import Poly, nextprime
%o from sympy.abc import x
%o def A106281_gen(): # generator of terms
%o p = 2
%o while True:
%o if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==5:
%o yield p
%o p = nextprime(p)
%o A106281_list = list(islice(A106281_gen(),20)) # _Chai Wah Wu_, Mar 14 2024
%Y 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 nonn
%O 1,1
%A _T. D. Noe_, May 02 2005
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