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%I #21 Aug 08 2023 04:45:45
%S 5,5,5,10,10,20,20,25,30,25,35,50,50,40,60,55,50,75,75,70,90,90,75,
%T 105,100,120,90,100,105,120,150,145,130,160,150,165,160,180,180,195,
%U 175,175,190,200,200,200,205,210,215,210,270,250,275,260,250,260,300
%N 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.
%C E is singular over GF(p(5)) = GF(11) so we take p != 11.
%C In other words, p runs through the primes other than 11.
%C 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.
%C Elkies proved that a(n) = prime(n+1) + 1 for infinitely many n.
%C 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.
%D N. Koblitz, Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.
%D J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.
%H Robin Visser, <a href="/A127310/b127310.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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>, Current Events Bulletin, Amer. Math. Soc., 2007.
%H B. Mazur, <a href="https://doi.org/10.1090/S0273-0979-08-01207-X">Finding meaning in error terms</a>, Bull. Amer. Math. Soc., 45 (No. 2, 2008), 185-228.
%F a(n) ~ prime(n+1) + 1 as n -> oo.
%F 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.
%F a(n) = 5*A127311(n).
%e 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.
%o (Sage)
%o def a(n):
%o if n < 5: p = Primes()[n-1]
%o else: p = Primes()[n]
%o E = EllipticCurve(GF(p), [0, -1, 1, 0, 0])
%o return E.cardinality() # _Robin Visser_, Jul 01 2023
%Y Cf. A000594, A127309, A127311.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Jan 12 2007
%E More terms from _Robin Visser_, Jul 01 2023
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