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Number of distinct zeros of x^2-x-1 mod prime(n).
7

%I #9 Apr 02 2015 17:32:38

%S 0,0,1,0,2,0,0,2,0,2,2,0,2,0,0,0,2,2,0,2,0,2,0,2,0,2,0,0,2,0,0,2,0,2,

%T 2,2,0,0,0,0,2,2,2,0,0,2,2,0,0,2,0,2,2,2,0,0,2,2,0,2,0,0,0,2,0,0,2,0,

%U 0,2,0,2,0,0,2,0,2,0,2,2,2,2,2,0,2,0,2,0,2,0,0,2,0,2,2,0,2,2,0,2,0,0,0,2,2

%N Number of distinct zeros of x^2-x-1 mod prime(n).

%C For the prime modulus 5, the polynomial can be factored as (x+2)^2, showing that x=3 is a zero of multiplicity 2. The discriminant of the polynomial is 5. Also note how this sequence is related to the Fibonacci sequence A051830; for n>3, a(n) = 2*A051830(n). - _T. D. Noe_, Aug 13 2004

%H J.-P. Serre, <a href="http://dx.doi.org/10.1090/S0273-0979-03-00992-3">On a theorem of Jordan</a>, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.

%F If p = prime(n), a(n) = A080891(p) + 1.

%t Table[p=Prime[n]; cnt=0; Do[If[Mod[x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 105}] (* _T. D. Noe_, Sep 24 2003 *)

%Y Cf. A086965, A086966, A086967.

%K nonn

%O 1,5

%A _N. J. A. Sloane_, Sep 23 2003

%E Corrected and extended by _T. D. Noe_, Sep 24 2003