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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
Union of A153636 and A154291. [From T. D. Noe (noe(AT)sspectra.com), Jan 06 2009]
Several numbers are formed in more than one way, e.g. 23, 31, 239, 499, 2687, 3299, 4027, 5323, 6079, ..., . [From Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 17 2009]
All terms have been checked using Sage. See A154291 for more details.
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REFERENCES
| 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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EXAMPLE
| 1427 = 4*-694^3 + 27*7037^2. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 17 2009]
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MATHEMATICA
| 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] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 17 2009]
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CROSSREFS
| Cf. A153636 (positive x only)
Sequence in context: A030670 A030680 A006203 * A052160 A165985 A093014
Adjacent sequences: A153632 A153633 A153634 * A153636 A153637 A153638
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Dec 29 2008, Jan 06 2009
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EXTENSIONS
| I added the Mathematica coding, extended the sequence - a(23)-a(45), added a Comment line and added an Example line. Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 17 2009
Comment corrected by T. D. Noe (noe(AT)sspectra.com), Jun 18 2009
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