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Search: id:A116570
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| A116570 |
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An all prime version of A114557 suggested by the Ono supersingular polynomials. |
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+0
1
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| 6, 3, 15, 5, 35, 7, 77, 11, 143, 13, 221, 17, 323, 19, 437, 23, 667, 29, 899, 31, 1147, 37, 1517, 41, 1763, 43, 2021, 47, 2491, 53, 3127, 59, 3599, 61, 4087, 67, 4757, 71, 5183, 73, 5767, 79, 6557, 83, 7387, 89, 8633, 97, 9797, 101, 10403, 103, 11021, 107, 11663
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OFFSET
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0,1
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COMMENT
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Mathematica for conjectured solution for supersingular S[x,n]: (* model polynomial*) s[x_, n_] = Mod[Expand[(x + Prime[n])*(x^2 + Prime[n + 1]*x + Prime[n + 1])], Prime[n + 2]] Table[s[x, n], {n, 9, 25}] (* get Weierstrass data*) p[n_] = Coefficient[s[x, n], x, 2] q[n_] = Coefficient[s[x, n], x, 1] r[n_] = Coefficient[s[x, n], x, 0] g2[n_] = -(3*q[n] - p[n]^2)/12 g3[n_] = -(2*p[n]^3 - 9*p[n]*q[n] + 27*r[n])/(27*4) j[n_] = FullSimplify[ExpandAll[g2[n]^3/(g2[n]^3 - 27*g3[n]^2)]] Table[{g2[n], g3[n], j[n]}, {n, 9, 25}] (*Only g3 isn't zero:*) g3[n]= -(1/4)*Mod[x^3 + x^2*Prime[n] + x* Prime[1 + n] + x^2*Prime[1 + n] + Prime[n]* Prime[1 + n] + x* Prime[n]*Prime[1 + n], Prime[2 + n]])
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REFERENCES
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Weierstrass points on X0(p) and supersingular j-invariants (Ken Ono with Scott Ahlgren) Mathematiche Annalen 325, 2003, pp. 355-368
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FORMULA
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Weierstrass elliptic form of a cubic: y^2 =4*x^3-g2[n]*x-g3[n]: a(n) = {g3[n]/4,g2[n]/4}
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MATHEMATICA
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a = Flatten[Table[Abs[Coefficient[Expand[(x +Prime[n])(x - Sqrt[Prime[n + 1]])*(x + Sqrt[Prime[n + 1]])], x, m]], {n, 1, 50}, {m, 0, 1}]]
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CROSSREFS
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Cf. A114557.
Sequence in context: A128756 A049784 A097917 this_sequence A046879 A067990 A050008
Adjacent sequences: A116567 A116568 A116569 this_sequence A116571 A116572 A116573
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KEYWORD
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nonn,uned,probation,obsc
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 18 2006
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