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A106279 Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 distinct zeros. 5
47, 53, 103, 163, 199, 257, 269, 311, 397, 401, 419, 421, 499, 587, 599, 617, 683, 757, 773, 863, 883, 907, 911, 929, 991, 1021, 1087, 1109, 1123, 1181, 1237, 1291, 1307, 1367, 1433, 1439, 1543, 1567, 1571, 1609, 1621, 1697, 1699, 1753, 1873, 1907, 2003 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
MATHEMATICA
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]]]
CROSSREFS
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)).
Sequence in context: A243431 A141279 A155139 * A275022 A355601 A048581
KEYWORD
nonn
AUTHOR
T. D. Noe, May 02 2005
STATUS
approved

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Last modified July 24 22:37 EDT 2024. Contains 374585 sequences. (Running on oeis4.)