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A127310
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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.
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2
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5, 5, 5, 10, 10, 20, 20, 25, 30, 25, 35, 50, 50, 40, 60, 55, 50, 75, 75
(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.
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) = p(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.
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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.
B. Mazur, Finding meaning in error terms, Bull. Amer. Math. Soc., 45 (No. 2, 2008), 185-228.
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) ~ p(n+1) + 1 as n -> oo.
a(n) = p+1 - 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 - 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.
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CROSSREFS
| a(n) = 5*A127311(n). Cf. A000594, A127309.
Sequence in context: A087516 A194428 A135089 * A101597 A119991 A131286
Adjacent sequences: A127307 A127308 A127309 * A127311 A127312 A127313
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KEYWORD
| nonn,more
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AUTHOR
| Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 12 2007
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