A127310 a(n) = |E(GF(p))| = number of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p where the prime p is p(n) or p(n+1) according as n < 5 or n >= 5.

5, 5, 5, 10, 10, 20, 20, 25, 30, 25, 35, 50, 50, 40, 60, 55, 50, 75, 75, 70, 90, 90, 75, 105, 100, 120, 90, 100, 105, 120, 150, 145, 130, 160, 150, 165, 160, 180, 180, 195, 175, 175, 190, 200, 200, 200, 205, 210, 215, 210, 270, 250, 275, 260, 250, 260, 300

Offset

1,1

Comments

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) = prime(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.

References

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

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

Links

Robin Visser, Table of n, a(n) for n = 1..10000

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, Current Events Bulletin, Amer. Math. Soc., 2007.

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

Formula

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

a(n) = p+1 - b(p) 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) = 5*A127311(n).

Example

q*Product_{k>=1} (1 - q^k)*(1 - q^(11*k))^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.

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()  # Robin Visser, Jul 01 2023

Crossrefs

Cf. A000594, A127309, A127311.

Sequence in context: A357331 A299695 A135089 * A214925 A101597 A119991

Adjacent sequences: A127307 A127308 A127309 * A127311 A127312 A127313

Keyword

nonn

Author

Jonathan Sondow, Jan 12 2007

Extensions

More terms from Robin Visser, Jul 01 2023

Status

approved