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A127309
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a(n) = |E(GF(p))| - (p+1) where E(GF(p)) is the group of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p and the prime p is p(n) or p(n+1) according as n < 5 or n >= 5.
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2
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2, 1, -1, 2, -4, 2, 0, 1, 0, -7, -3, 8, 6, -8, 6, -5, -12, 7, 3
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| E is singular over GF(p(5)) = GF(11) so we take p != 11.
Hasse proved that |a(n)| <= 2*sqrt(p) where p is p(n) or p(n+1) according as n < 5 or n >= 5.
Elkies proved that a(n) = 0 for infinitely many n.
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REFERENCES
| 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.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986.
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LINKS
| 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
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FORMULA
| a(n) = -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.
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EXAMPLE
| q*Prod(k=1 to oo, ((1 - q^k)(1 - q^11k))^2) = q - 2q^2 - ..., so a(1) = -b(p(1)) = -b(2) = -(-2) = 2.
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CROSSREFS
| |E(GF(p))| is A127310. Cf. A000594, A127311.
Sequence in context: A056061 A029265 A103648 * A097853 A160266 A023504
Adjacent sequences: A127306 A127307 A127308 * A127310 A127311 A127312
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KEYWORD
| sign
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AUTHOR
| Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 12 2007
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