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A272202
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Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.
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2
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2, 3, 5, 3, 11, 11, 17, 27, 23, 29, 27, 47, 41, 51, 47, 53, 59, 47, 51, 71, 83, 75, 83, 89, 83, 101, 123, 107, 107, 113, 147, 131, 137, 123, 149, 147, 143, 171, 167, 173, 179, 155, 191, 191, 197, 171, 195, 195, 227, 251, 233, 239, 227, 251, 257, 263, 269, 243, 251, 281
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OFFSET
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1,1
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COMMENTS
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In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the last column, starting with Conductor 144, as a strong Weil curve for the weight 2 newform eta^{12}(12*z) / (eta^4(6*z) * eta^4(24*z)), symbolically 12^{12} 6^{-4} 24^{-4}, with Im(z) > 0, and the Dedekind eta function. See A187076 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)* eta(2*z)^{12} / (eta^4(z)*eta^4(4*z)). For the q-expansion of 12^{12} 6^{-4} 24^{-4} one has a leading zero and 5 interspersed 0's: 0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,-8,...
The discriminant of this elliptic curve is -3^3 = -27.
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LINKS
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FORMULA
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a(n) gives the number of solutions of the congruence y^2 == x^3 - 1 (mod prime(n)), n >= 1.
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EXAMPLE
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The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 1 (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1, 2, 2: (0, 1), (1, 0)
2, 3, 3: (1, 0), (2, 1), (2, 2)
3, 5, 5: (0, 2), (0, 3), (1, 0),
(3, 1), (3, 4)
4, 7, 3: (1, 0), (2, 0), (4, 0)
5, 11, 11: (1, 0), (3, 2), (3, 9),
(5, 5), (5, 6), (7, 1),
(7, 10), (8, 4), (8, 7),
(10, 3), (10, 8)
...
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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