%I A127309
%S A127309 2,1,1,2,4,2,0,1,0,7,3,8,6,8,6,5,12,7,3
%V A127309 2,1,-1,2,-4,2,0,1,0,-7,-3,8,6,-8,6,-5,-12,7,3
%N A127309 a(n) = |E(GF(p))| - (p+1) where E(GF(p)) is the group of rational points 
               on the elliptic curve E: y^2 + y = x^3 - x^2 mod p and the prime 
               p is p(n) or p(n+1) according as n < 5 or n >= 5.
%C A127309 E is singular over GF(p(5)) = GF(11) so we take p != 11.
%C A127309 Hasse proved that |a(n)| <= 2*sqrt(p) where p is p(n) or p(n+1) according 
               as n < 5 or n >= 5.
%C A127309 Elkies proved that a(n) = 0 for infinitely many n.
%D A127309 N. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: 
               Springer-Verlag, 1993.
%D A127309 B. Mazur, The Structure of Error Terms in Number Theory and an Introduction 
               to the Sato-Tate Conjecture, Current Events Bulletin, Amer. Math. 
               Soc., 2007.
%D A127309 J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in 
               Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.
%H A127309 S. Fermigier, <a href="http://www.fermigier.com/fermigier/elliptic.html.en">
               Collection of Links on Research Articles on Elliptic Curves and Related 
               Topics</a>
%H A127309 B. Mazur, <a href="http://www.ams.org/ams/currentevents2007.pdf">The 
               Structure of Error Terms in Number Theory and an Introduction to 
               the Sato-Tate Conjecture</a>
%F A127309 a(n) = -b(p) where q*Prod(k=1 to oo, ((1 - q^k)(1 - q^11k))^2) = Sum(k=1 
               to oo, b(k)*q^k) and p is p(n) or p(n+1) according as n < 5 or n 
               >= 5.
%e A127309 q*Prod(k=1 to oo, ((1 - q^k)(1 - q^11k))^2) = q - 2q^2 - ..., so a(1) 
               = -b(p(1)) = -b(2) = -(-2) = 2.
%Y A127309 |E(GF(p))| is A127310. Cf. A000594, A127311.
%Y A127309 Sequence in context: A029265 A103648 A133771 this_sequence A097853 A160266 
               A023504
%Y A127309 Adjacent sequences: A127306 A127307 A127308 this_sequence A127310 A127311 
               A127312
%K A127309 sign
%O A127309 1,1
%A A127309 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 12 2007