

A098513


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


2



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
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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^(k1) for k>1 and, for odd primes p, a(p^k) = a(p) p^(k1)


MATHEMATICA

Table[s2=Mod[Table[y^2, {y, 0, n1}], n]; s3=Mod[Table[x^3+x+1, {x, 0, n1}], 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



