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A199263
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Let CardE(K) the number of elements (x, y) of the finite group defined by the elliptic curve y^2 = x^3 + x + 1 (mod p) including the point at infinity ; a(n) is the difference between 2*sqrt(p) and |cardE(K) - (p + 1)|.
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2
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2, 3, 1, 2, 4, 3, 8, 7, 5, 4, 10, 2, 5, 3, 1, 10, 12, 3, 4, 3, 15, 11, 12, 8, 18, 17, 3, 17, 7, 10, 20, 18, 11, 9, 10, 22, 12, 0, 1, 24, 26, 18, 2, 20, 4, 10, 18, 9, 30, 28, 27, 8, 9, 1, 23, 28, 8, 30, 11, 26, 13, 8, 28, 32, 3, 20, 26, 22, 21, 23, 33, 28, 16
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OFFSET
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1,1
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COMMENTS
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Hasse’s theorem tells us the following estimate :
|cardE(K) - (p + 1)|<= 2*sqrt(p) where K = Fp is the finite field with p elements and E an elliptic curve y^2 = x^3 + x + 1 (mod p) defined over K. CardE(K) is the number of elements (x, y) of the finite group defined by the elliptic curve including the point at infinity. The number of points of the curve grows roughly as the number of elements in the field. This sequence gives the integer difference : 2*sqrt(p) - |cardE(K) - (p + 1)|.
We obtain remarkable values such that a(38) = 0, a(258) = 0.
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LINKS
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Michel Lagneau, Table of n, a(n) for n = 1..1000
Wolfram MathWorld, , Elliptic Curve Group Law.
Wikipedia, Hasse's theorem on elliptic curves
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EXAMPLE
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For n=6, p = prime(6)= 13 and a(6) = 3 because the solutions of y^2 = x^3 + x + 1 (mod 13) are {(inf, inf), (7,0), (0,1), (5, 1), (8,1), (4,2), (11,2), (1, 4), (12,5), (10,6), (10,7), (12, 8), (1,9), (4,11), (11,11), (0, 12), (5,12), (8,12)} => CardE(K) = A192334(6) = 18, and floor(2*sqrt(13) - 18 + 13 + 1) = floor(7.2111025 -4) = 3.
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MAPLE
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for m from 1 to 100 do:p:=ithprime(m):it:=1:for y from 0 to p-1 do:for x from 0 to p-1 do:z:=x^3+x+1:z1:=irem(z, p):z2:=irem(y^2, p):if z1=z2 then it:=it+1:else fi:od:od:y:= floor(2*evalf(sqrt(p))-abs(it-p-1)): printf(`%d, `, y):od:
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CROSSREFS
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Cf. A192334, A098514.
Sequence in context: A035459 A048232 A163256 * A181803 A144962 A166871
Adjacent sequences: A199260 A199261 A199262 * A199264 A199265 A199266
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KEYWORD
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nonn
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AUTHOR
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Michel Lagneau, Nov 07 2011
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STATUS
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approved
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