A086937 Number of distinct zeros of x^2-x-1 mod prime(n).

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Sep 23 2003

Keywords

• nonn

Comments

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

Links

• J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.

Crossrefs

Cf. A086965, A086966, A086967.

Programs

Mathematica

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 *)

Formula

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

Extensions

Corrected and extended by T. D. Noe, Sep 24 2003