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A106277
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Number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n).
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3
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0, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 2, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 2, 2, 2, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 2, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 2, 1, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,10
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COMMENTS
| This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -563 and 563 is the only prime for which the polynomial has 3 distinct zeros. The primes p yielding 4 distinct zeros, A106280, correspond to the periods of the sequences A000078(k) mod p and A073817(k) mod p having length less than p. The Lucas 4-step sequence mod p has one additional prime p for which the period is less than p: the discriminant 563. For this prime, the Fibonacci 4-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^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]
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CROSSREFS
| Cf. A106296 (period of the Lucas 4-step sequences mod prime(n)), A106283 (prime moduli for which the polynomial is irreducible).
Sequence in context: A151758 A164272 A164273 * A088627 A024713 A123530
Adjacent sequences: A106274 A106275 A106276 * A106278 A106279 A106280
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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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