A086966 Number of distinct zeros of x^4-x-1 mod prime(n).

0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 0, 2, 1, 0, 0, 2, 1, 1, 2, 0, 0, 2, 4, 1, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 2, 2, 2, 1, 1, 0, 0, 2, 1, 2, 2, 1, 1, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 0, 2, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 2, 1, 1, 0

Offset: 1

Author

N. J. A. Sloane, Sep 24 2003

Keywords

• nonn

Comments

For the prime modulus 283, the polynomial can be factored as (x+18) (x+168) (x+190)^2, showing that x=93 is a zero of multiplicity 2. The discriminant of the polynomial is 283. - T. D. Noe, Aug 12 2004

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see pp. 433-434.

Crossrefs

Cf. A086937, A086965, A086967.

Programs

Maple

f:= n -> nops([msolve(x^4-x-1,ithprime(n))]):
map(f, [$1..100]); # Robert Israel, Aug 10 2023

Mathematica

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 105}] (* T. D. Noe, Sep 24 2003 *)

Extensions

More terms from T. D. Noe, Sep 24 2003