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Search: id:A098514
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| A098514 |
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For p=prime(n), a(n) = number of points (x,y) on the elliptic curve y^2 = x^3 + x + 1 (mod p), not including the point at infinity. |
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+0
3
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| 2, 3, 8, 4, 13, 17, 17, 20, 27, 35, 32, 47, 34, 33, 59, 57, 62, 49, 55, 58, 71, 85, 89, 99, 96, 104, 86, 104, 122, 124, 125, 127, 125, 125, 135, 153, 170, 188, 143, 171, 179, 189, 216, 200, 221, 217, 222, 243, 227, 231, 236, 261, 219, 281, 248, 259, 293, 273, 255, 288
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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This is one of the simplest non-degenerate elliptic curves. A theorem of Hasse states that the number of points (including the point at infinity) is p+1+d, where |d| < 2 sqrt(p).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Joseph H. Silverman, The Ubiquity of Elliptic Curves (PowerPoint)
Eric Weisstein's World of Mathematics, Elliptic Curve
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MATHEMATICA
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Table[p=Prime[n]; s2=Mod[Table[y^2, {y, 0, p-1}], p]; s3=Mod[Table[x^3+x+1, {x, 0, p-1}], p]; s=Intersection[Union[s2], Union[s3]]; Sum[Count[s2, s[[i]]]*Count[s3, s[[i]]], {i, Length[s]}], {n, 100}]
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CROSSREFS
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Cf. A098513 (number of points on the elliptic curve y^2 = x^3 + x + 1 (mod n)).
Sequence in context: A100869 A110142 A158928 this_sequence A161198 A093898 A084110
Adjacent sequences: A098511 A098512 A098513 this_sequence A098515 A098516 A098517
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Sep 11 2004
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