1,1

E is singular over GF(p(5)) = GF(11) so we take p != 11.

In other words, p runs through the primes other than 11.

Hasse proved that |a(n) - (p+1)| <= 2*sqrt(p) where p is p(n) or p(n+1) according as n < 5 or n >= 5.

Elkies proved that a(n) = p(n+1) + 1 for infinitely many n.

a(n) is divisible by 5 because the points oo, (0,0), (0,-1), (1,0), (1,-1) on E form a subgroup of E(GF(p)) of order 5.

N. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.

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.

B. Mazur, Finding meaning in error terms, Bull. Amer. Math. Soc., 45 (No. 2, 2008), 185-228.

J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.

S. Fermigier, Collection of Links on Research Articles on Elliptic Curves and Related Topics

B. Mazur, The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture

a(n) ~ p(n+1) + 1 as n -> oo.

a(n) = p+1 - 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.

q*Prod(k=1 to oo, ((1 - q^k)(1 - q^11k))^2) = q - 2q^2 - q^3 + ..., so a(1) = p(1) + 1 - b(p(1)) = 2 + 1 - b(2) = 3 - (-2) = 5 and a(2) = p(2) + 1 - b(p(2)) = 3 + 1 - b(3) = 4 - (-1) = 5.

a(n) = 5*A127311(n). Cf. A000594, A127309.

Sequence in context: A127934 A087516 A135089 this_sequence A101597 A119991 A131286

Adjacent sequences: A127307 A127308 A127309 this_sequence A127311 A127312 A127313

nonn,more

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 12 2007