Search: id:A153635
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%I A153635
%S A153635 23,31,59,139,211,239,283,419,491,499,563,643,743,751,823,1291,1319,
%T A153635 1327,1399,1427,1579,1823,1931,2039,2687,2767,3011,3119,3163,3191,3271,
%U A153635 3299,3307,3371,3559,3767,3803,3919,4027,4091,4099,4243,4423,4567,4639
%N A153635 Primes of the form 4x^3 + 27y^2, with x positive or negative.
%C A153635 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.
%C A153635 Union of A153636 and A154291. [From T. D. Noe (noe(AT)sspectra.com), 
               Jan 06 2009]
%C A153635 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]
%C A153635 All terms have been checked using Sage. See A154291 for more details.
%D A153635 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               6th Edition, Oxford Univ. Press, 2008, p. 595.
%e A153635 1427 = 4*-694^3 + 27*7037^2. [From Robert G. Wilson v (rgwv(AT)rgwv.com), 
               Feb 17 2009]
%t A153635 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]
%Y A153635 Cf. A153636 (positive x only)
%Y A153635 Sequence in context: A030670 A030680 A006203 this_sequence A052160 A165985 
               A093014
%Y A153635 Adjacent sequences: A153632 A153633 A153634 this_sequence A153636 A153637 
               A153638
%K A153635 nonn
%O A153635 1,1
%A A153635 T. D. Noe (noe(AT)sspectra.com), Dec 29 2008, Jan 06 2009
%E A153635 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
%E A153635 Comment corrected by T. D. Noe (noe(AT)sspectra.com), Jun 18 2009

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