OFFSET
1,4
COMMENTS
H is a subgroup of E(GF(p)) of order 5 so a(n) = |E(GF(p))|/5 where p is p(n) or p(n+1) according as n < 5 or n >= 5.
E is singular over GF(p(5)) = GF(11) so we take p != 11.
Hasse proved that |5*a(n) - (p+1)| <= 2*sqrt(p) where p is prime(n) or prime(n+1) according as n < 5 or n >= 5.
Elkies proved that 5*a(n) = prime(n+1) + 1 for infinitely many n.
REFERENCES
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.
LINKS
Robin Visser, Table of n, a(n) for n = 1..10000
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.
FORMULA
a(n) ~ (prime(n+1) + 1)/5 as n -> oo.
a(n) = (p+1 - b(p))/5 where q*Product_{k>=1} (1 - q^k)*(1 - q^(11*k))^2 = Sum_{k>=1} b(k)*q^k and p is prime(n) or prime(n+1) according as n < 5 or n >= 5.
a(n) = A127310(n)/5.
EXAMPLE
q*Prod(k=1 to oo, ((1 - q^k)(1 - q^11k))^2) = q - 2q^2 - ..., so a(1) = (p(1) + 1 - b(p(1))/5 = (2 + 1 - b(2))/5 = (3 - (-2))/5 = 1.
PROG
(Sage)
def a(n):
if n < 5: p = Primes()[n-1]
else: p = Primes()[n]
E = EllipticCurve(GF(p), [0, -1, 1, 0, 0])
return E.cardinality()/5 # Robin Visser, Jul 01 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 12 2007
EXTENSIONS
More terms from Robin Visser, Jul 01 2023
STATUS
approved