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A106276
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Number of distinct zeros of x^3-x^2-x-1 mod prime(n).
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4
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1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 3, 3, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 3, 1, 3, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 3, 3, 1, 3, 3, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step recursions, A000073 and A001644. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -44 = -4*11. The primes p yielding 3 distinct zeros, A106279, correspond to the periods of the sequences A000073(k) mod p and A001644(k) mod p having length less than p. The Lucas 3-step sequence mod p has two additional primes p for which the period is less than p: 2 and 11, which are factors of the discriminant -44. For p=11, the Fibonacci 3-step sequence mod p has a period of p(p-1).
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LINKS
| J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
Eric Weisstein's World of Mathematics, Fibonacci n-Step
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MATHEMATICA
| Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]
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CROSSREFS
| Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106293 (period of the Lucas 3-step sequences mod prime(n)), A106282 (prime moduli for which the polynomial is irreducible).
Sequence in context: A072617 A056226 A044935 * A037907 A037801 A053252
Adjacent sequences: A106273 A106274 A106275 * A106277 A106278 A106279
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), May 02 2005
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