A192334 Number of elements of the finite group defined by the elliptic curve y^2 = x^3 + x + 1 (mod prime(n)).

3, 4, 9, 5, 14, 18, 18, 21, 28, 36, 33, 48, 35, 34, 60, 58, 63, 50, 56, 59, 72, 86, 90, 100, 97, 105, 87, 105, 123, 125, 126, 128, 126, 126, 136, 154, 171, 189, 144, 172, 180, 190, 217, 201, 222, 218, 223, 244, 228, 232, 237, 262, 220, 282, 249, 260, 294, 274, 256, 289

Offset

1,1

Comments

Any elliptic curve can be written as a plane algebraic curve defined by an equation of the form: y^2 = x^3 + ax + b.

Links

Robin Visser, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Elliptic Curve Group Law.

Formula

a(n) = 1+A098514(n), which does not count (oo,oo). - R. J. Mathar, Jun 29 2011

Example

Example : a(8) = 21 because, for n = 8, prime(8) = 19, and the field F(19) has 21 elements :(19, 1), (10, 2), (14, 2), (7, 3), (15, 3), (16, 3), (5, 6), (9, 6), (2, 7), (13, 8), (13, 11), (2, 12), (5, 13), (9, 13), (7, 16), (15, 16), (16, 16), (10, 17), (14, 17), (19,18) and the point at infinity.

Maple

with(numtheory):for n from 1 to 100 do:p:=ithprime(n):it:=0:for y from 1 to p do:for x from 1 to p 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: printf(`%d, `, it+1): od:

Mathematica

Reap[ For[ n=1, n <= 60 , n++, p = Prime[n]; it=0; For[ y=1 , y <= p , y++, For[ x=1 , x <= p , x++, z = x^3+x+1; z1 = Mod[z, p]; z2 = Mod[y^2, p]; If[ z1 == z2 , it = it+1]]]; Sow[it+1]]][[2, 1]](* Jean-François Alcover, Jun 11 2012, translated from Maple *)

Crossrefs

Sequence in context: A330403 A183211 A256469 * A140439 A023183 A102320

Adjacent sequences: A192331 A192332 A192333 * A192335 A192336 A192337

Keyword

nonn

Author

Michel Lagneau, Jun 28 2011

Status

approved