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A304293
Number of points of a Koblitz curve E: y^2 + x*y = x^3 + a*x^2 + 1 over a field with 2^n elements.
1
0, 4, 8, 4, 16, 44, 56, 116, 288, 508, 968, 2116, 4144, 8012, 16472, 33044, 65088, 130972, 263144, 523492, 1047376, 2099948, 4193912, 8383412, 16783200, 33558844, 67092488, 134225284, 268460656, 536830604, 1073731736, 2147574356, 4294896768, 8589823708
OFFSET
0,2
COMMENTS
A general result of Bilu, Hong, & Luca proves that, for n > e^e^10^10, a(n) has a prime divisor p > n*exp(0.0001*log n/log log n). - Charles R Greathouse IV, Feb 14 2022
LINKS
Yuri Bilu, Haojie Hong, and Florian Luca, Big prime factors in orders of elliptic curves over finite fields, arXiv:2112.07046 [math.NT], 2021.
FORMULA
G.f.: (4*x - 8*x^3) / (1 - 2*x + x^2 - 4*x^3 + 4*x^4).
a(n) = 2^n + 1 - ((-1 + 7 i)/2)^n - ((-1 - 7 i)/2)^n.
a(n) = a(-n) * 2^n for all n in Z.
EXAMPLE
G.f. = 4*x + 8*x^2 + 4*x^3 + 16*x^4 + 44*x^5 + 56*x^6 + 116*x^7 + ...
MATHEMATICA
a[ n_] := Simplify[ 2^n + 1 - ((-1 + Sqrt[-7]) / 2)^n - ((-1 - Sqrt[-7]) / 2)^n];
CoefficientList[Series[(4*x-8*x^3)/(1-2*x+x^2-4*x^3+4*x^4), {x, 0, 50}], x] (* G. C. Greubel, Jul 28 2018 *)
PROG
(PARI) {a(n) = my(w=-quadgen(-7)); simplify(2^n + 1 - w^n - (-1-w)^n)};
(PARI) x='x+O('x^30); concat([0], Vec((4*x-8*x^3)/(1-2*x+x^2-4*x^3+ 4*x^4))) \\ G. C. Greubel, Jul 28 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((4*x -8*x^3)/(1-2*x+x^2-4*x^3+4*x^4))); // G. C. Greubel, Jul 28 2018
CROSSREFS
Sequence in context: A028587 A087260 A019254 * A055374 A275876 A255293
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jun 06 2018
STATUS
approved