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

%I #17 Aug 12 2023 00:42:52

%S 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,

%T 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,

%U 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

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

%C 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

%H Robert Israel, <a href="/A086966/b086966.txt">Table of n, a(n) for n = 1..10000</a>

%H J.-P. Serre, <a href="https://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 pp. 433-434.

%p f:= n -> nops([msolve(x^4-x-1,ithprime(n))]):

%p map(f, [$1..100]); # _Robert Israel_, Aug 10 2023

%t 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 *)

%Y Cf. A086937, A086965, A086967.

%K nonn

%O 1,7

%A _N. J. A. Sloane_, Sep 24 2003

%E More terms from _T. D. Noe_, Sep 24 2003