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, 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

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

Eric Weisstein's World of Mathematics, 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 * A341826 A372479 A257669

Adjacent sequences: A199260 A199261 A199262 * A199264 A199265 A199266

Keyword

nonn

Author

Michel Lagneau, Nov 07 2011

Status

approved