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A199263 Let CardE(K) the number of elements (x, y) of the finite group defined by the elliptic curve y^2 = x^3 + x + 1 (mod p) including the point at infinity ; a(n) is the difference between 2*sqrt(p) and  |cardE(K) - (p + 1)|. 2
2, 3, 1, 2, 4, 3, 8, 7, 5, 4, 10, 2, 5, 3, 1, 10, 12, 3, 4, 3, 15, 11, 12, 8, 18, 17, 3, 17, 7, 10, 20, 18, 11, 9, 10, 22, 12, 0, 1, 24, 26, 18, 2, 20, 4, 10, 18, 9, 30, 28, 27, 8, 9, 1, 23, 28, 8, 30, 11, 26, 13, 8, 28, 32, 3, 20, 26, 22, 21, 23, 33, 28, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Hasse’s theorem tells us the following estimate :

|cardE(K) - (p + 1)|<= 2*sqrt(p) where K = Fp is the finite field with p elements and E an elliptic curve y^2 = x^3 + x + 1 (mod p) defined over K. CardE(K) is the number of elements (x, y) of the finite group defined by the elliptic curve  including the point at infinity. The number of points of the curve grows roughly as the number of elements in the field. This sequence gives the integer difference  : 2*sqrt(p) - |cardE(K) - (p + 1)|.

We obtain remarkable values such that a(38) = 0, a(258) = 0.

LINKS

Michel Lagneau, Table of n, a(n) for n = 1..1000

Wolfram MathWorld, , Elliptic Curve Group Law.

Wikipedia, Hasse's theorem on elliptic curves

EXAMPLE

For n=6, p = prime(6)= 13 and a(6) = 3  because the  solutions of y^2 = x^3 + x + 1 (mod 13) are {(inf, inf), (7,0), (0,1), (5, 1), (8,1), (4,2), (11,2), (1, 4), (12,5), (10,6), (10,7), (12, 8), (1,9), (4,11), (11,11), (0, 12), (5,12), (8,12)} => CardE(K) = A192334(6) = 18, and floor(2*sqrt(13) - 18 + 13 + 1) = floor(7.2111025 -4) = 3.

MAPLE

for m from 1 to 100 do:p:=ithprime(m):it:=1:for y from 0 to p-1 do:for x from 0 to p-1 do:z:=x^3+x+1:z1:=irem(z, p):z2:=irem(y^2, p):if z1=z2 then it:=it+1:else fi:od:od:y:=  floor(2*evalf(sqrt(p))-abs(it-p-1)): printf(`%d, `, y):od:

CROSSREFS

Cf. A192334, A098514.

Sequence in context: A035459 A048232 A163256 * A181803 A144962 A227542

Adjacent sequences:  A199260 A199261 A199262 * A199264 A199265 A199266

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 07 2011

STATUS

approved

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Last modified July 28 00:10 EDT 2014. Contains 244987 sequences.