1,2

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.

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

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

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}]

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

mult,nice,nonn

T. D. Noe, Sep 11 2004

approved