A098514 For p=prime(n), a(n) = number of points (x,y) on the elliptic curve y^2 = x^3 + x + 1 (mod p), not including the point at infinity.

2, 3, 8, 4, 13, 17, 17, 20, 27, 35, 32, 47, 34, 33, 59, 57, 62, 49, 55, 58, 71, 85, 89, 99, 96, 104, 86, 104, 122, 124, 125, 127, 125, 125, 135, 153, 170, 188, 143, 171, 179, 189, 216, 200, 221, 217, 222, 243, 227, 231, 236, 261, 219, 281, 248, 259, 293, 273, 255, 288

Offset

1,1

Comments

This is one of the simplest non-degenerate elliptic curves. A theorem of Hasse states that the number of points (including the point at infinity) is p+1+d, where |d| < 2 sqrt(p).

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Joseph H. Silverman, The Ubiquity of Elliptic Curves (PowerPoint)

Eric Weisstein's World of Mathematics, Elliptic Curve

Mathematica

Table[p=Prime[n]; s2=Mod[Table[y^2, {y, 0, p-1}], p]; s3=Mod[Table[x^3+x+1, {x, 0, p-1}], p]; s=Intersection[Union[s2], Union[s3]]; Sum[Count[s2, s[[i]]]*Count[s3, s[[i]]], {i, Length[s]}], {n, 100}]

Crossrefs

Cf. A098513 (number of points on the elliptic curve y^2 = x^3 + x + 1 (mod n)).

Sequence in context: A193731 A193975 A224665 * A161198 A195232 A093898

Adjacent sequences: A098511 A098512 A098513 * A098515 A098516 A098517

Keyword

nonn

Author

T. D. Noe, Sep 11 2004

Status

approved