A098513 Number of points (x,y) on the elliptic curve y^2 = x^3 + x + 1 (mod n), not including the point at infinity.

1, 2, 3, 2, 8, 6, 4, 4, 9, 16, 13, 6, 17, 8, 24, 8, 17, 18, 20, 16, 12, 26, 27, 12, 40, 34, 27, 8, 35, 48, 32, 16, 39, 34, 32, 18, 47, 40, 51, 32, 34, 24, 33, 26, 72, 54, 59, 24, 28, 80, 51, 34, 57, 54, 104, 16, 60, 70, 62, 48, 49, 64, 36, 32, 136, 78, 55, 34, 81, 64, 58, 36, 71, 94

Offset

1,2

Comments

This is one of the simplest nondegenerate elliptic curves. By not including the point at infinity, we can see the multiplicative structure of this sequence. A theorem of Hasse states that, for prime n, the number of points (including the point at infinity) is n+1+d, where |d| < 2 sqrt(n). When a(n) is an odd prime, then n is prime.

Links

T. D. Noe, Table of n, a(n) for n=1..2000

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

Eric Weisstein's World of Mathematics, Elliptic Curve

Formula

multiplicative rule: a(2^k) = 2^(k-1) for k>1 and, for odd primes p, a(p^k) = a(p) p^(k-1)

Mathematica

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

Crossrefs

Cf. A098514 (number of points on the elliptic curve y^2 = x^3 + x + 1 (mod prime(n))).

Sequence in context: A136193 A187789 A245922 * A134347 A057761 A321477

Adjacent sequences: A098510 A098511 A098512 * A098514 A098515 A098516

Keyword

mult,nice,nonn

Author

T. D. Noe, Sep 11 2004

Status

approved