

A106278


Number of distinct zeros of x^5x^4x^3x^2x1 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
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OFFSET

1,9


COMMENTS

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5step 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 5step 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 5step sequence mod p has a period of p(p1).


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), 429440, see p. 433.
Eric Weisstein's World of Mathematics, Fibonacci nStep


MATHEMATICA

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^5x^4x^3x^2x1, p]==0, cnt++ ], {x, 0, p1}]; cnt, {n, 150}]


CROSSREFS

Cf. A106298 (period of the Lucas 5step 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



