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%I #12 Mar 24 2024 07:55:25
%S 47,53,103,163,199,257,269,311,397,401,419,421,499,587,599,617,683,
%T 757,773,863,883,907,911,929,991,1021,1087,1109,1123,1181,1237,1291,
%U 1307,1367,1433,1439,1543,1567,1571,1609,1621,1697,1699,1753,1873,1907,2003
%N Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 distinct zeros.
%C This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644. The periods of the sequences A000073(k) mod p and A001644(k) mod p have length less than p. For a given p, let the zeros be a, b and c. Then A001644(k) mod p = (a^k+b^k+c^k) mod p. This sequence is the same as A033209 except for the initial term.
%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^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 500}];Prime[Flatten[Position[t, 3]]]
%Y Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (periods of the Fibonacci and Lucas 3-step sequences mod prime(n)).
%K nonn
%O 1,1
%A _T. D. Noe_, May 02 2005