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A106278 Number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n). 3
1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 2, 3, 0, 2, 3, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 3, 1, 2, 3, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 1, 0, 0, 0, 1, 1, 1, 2, 1, 2, 0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 2, 2, 1, 1, 2, 0, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is 9584=16*599 and 599 is the only prime for which the polynomial has 4 distinct zeros. The primes p yielding 5 distinct zeros, A106281, correspond to the periods of the sequences A001591(k) mod p and A074048(k) mod p having length less than p. The Lucas 5-step sequence mod p has one additional prime p for which the period is less than p: the 599 factor of the discriminant. For this prime, the Fibonacci 5-step sequence mod p has a period of p(p-1).

LINKS

Table of n, a(n) for n=1..105.

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

MATHEMATICA

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, 150}]

CROSSREFS

Cf. A106298 (period of the Lucas 5-step sequences mod prime(n)), A106284 (prime moduli for which the polynomial is irreducible).

Sequence in context: A324351 A116402 A093323 * A177517 A227819 A064287

Adjacent sequences:  A106275 A106276 A106277 * A106279 A106280 A106281

KEYWORD

nonn

AUTHOR

T. D. Noe, May 02 2005

STATUS

approved

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Last modified August 11 23:45 EDT 2020. Contains 336434 sequences. (Running on oeis4.)