A098513 - OEIS

%I #7 Mar 30 2012 17:22:33

%S 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,

%T 8,35,48,32,16,39,34,32,18,47,40,51,32,34,24,33,26,72,54,59,24,28,80,

%U 51,34,57,54,104,16,60,70,62,48,49,64,36,32,136,78,55,34,81,64,58,36,71,94

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

%C 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.

%H T. D. Noe, <a href="/A098513/b098513.txt">Table of n, a(n) for n=1..2000</a>

%H Joseph H. Silverman, <a href="http://www.math.brown.edu/~jhs/UbiquityOfEllipticCurves.ppt">The Ubiquity of Elliptic Curves (PowerPoint)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipticCurve.html">Elliptic Curve</a>

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

%t 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}]

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

%K mult,nice,nonn

%O 1,2

%A _T. D. Noe_, Sep 11 2004