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A127311 a(n) = |E(GF(p))/H| where E(GF(p)) is the group of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p, the prime p is p(n) or p(n+1) according as n < 5 or n >= 5 and H = {oo, (0,0), (0,-1), (1,0), (1,-1)}. 2
1, 1, 1, 2, 2, 4, 4, 5, 6, 5, 7, 10, 10, 8, 12, 11, 10, 15, 15 (list; graph; refs; listen; history; text; internal format)
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 p(n) or p(n+1) according as n < 5 or n >= 5.

Elkies proved that 5*a(n) = p(n+1) + 1 for infinitely many n.

REFERENCES

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.

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

Table of n, a(n) for n=1..19.

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

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

FORMULA

a(n) ~ (p(n+1) + 1)/5 as n -> oo.

a(n) = (p+1 - b(p))/5 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.

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.

CROSSREFS

a(n) = A127310(n)/5. Cf. A000594, A127309.

Sequence in context: A113474 A089413 A159267 * A302929 A129229 A219029

Adjacent sequences:  A127308 A127309 A127310 * A127312 A127313 A127314

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Jan 12 2007

STATUS

approved

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Last modified October 1 00:23 EDT 2020. Contains 337440 sequences. (Running on oeis4.)