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A153635
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Primes of the form 4x^3 + 27y^2, with x positive or negative.
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2
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23, 31, 59, 139, 211, 239, 283, 419, 491, 499, 563, 643, 743, 751, 823, 1291, 1319, 1327, 1399, 1427, 1579, 1823, 1931, 2039, 2687, 2767, 3011, 3119, 3163, 3191, 3271, 3299, 3307, 3371, 3559, 3767, 3803, 3919, 4027, 4091, 4099, 4243, 4423, 4567, 4639
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OFFSET
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1,1
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COMMENTS
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Hardy and Wright: If there are an infinite number of these primes, then there are infinitely many cubic polynomials with integer coefficients and prime discriminant. It would also resolve the open conjecture that there are infinitely many non-isomorphic elliptic curves defined over the rationals and having prime conductor.
Several numbers are formed in more than one way, e.g. 23, 31, 239, 499, 2687, 3299, 4027, 5323, 6079, ..., . - Robert G. Wilson v, Feb 17 2009
All terms have been checked using Sage. See A154291 for more details.
Granville: "The most desired open problem in [asymptotic sieves] is to show that 4a^3 + 27b^2 is prime for infinitely many pairs of integers a, b (this is of interest because if 4a^3 + 27b^2 is prime then it is usually the conductor of the elliptic curve y^2 = x^3 + ax + b)." - Charles R Greathouse IV, Jun 06 2013
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition, Oxford Univ. Press, 2008, p. 595.
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LINKS
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EXAMPLE
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MATHEMATICA
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lst = {}; Do[ If[ z = 4x^3 + 27y^2; 0 < z < 10000 && PrimeQ@z, AppendTo[lst, z]; Print[{z, x, y}]], {y, 25000}, {x, -Floor[(27 y^2/4)^(1/3)], -Floor[(27 y^2/4)^(1/3)] + 100}]; Take[ Union@ lst, 45] (* Robert G. Wilson v, Feb 17 2009 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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