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%I #9 Oct 05 2013 12:41:18
%S 3,5,32,35,37,63,65,136,568,5215,367806,939787,6369039,7885438,
%T 9536129,140292677,184151166,890838663,912903445,3171881612
%N Values x for "records of minima" of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.
%C "Records of minima" means values A154333(x) such that A154333(x') > A154333(x) for all x' > x. See the main entry A179386 for all further considerations. - _M. F. Hasler_, Sep 30 2013
%C For d values see A179386; For y values see A179388.
%C Theorem (Artur Jasinski):
%C For any positive number x >= A179387(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n).
%C Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and complete computable can't existed such x.
%C From _Artur Jasinski_, Aug 11 2010: (Start)
%C An equivalent theorem is the following (Artur Jasinski):
%C For any positive number x >= 1+A179387(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n+1).
%C (End)
%t max = 1000; vecd = Table[10100, {n, 1, max}]; vecx = Table[10100, {n, 1, max}]; vecy = Table[10100, {n, 1, max}]; len = 1; min = 10100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx (*Artur Jasinski*)
%Y Cf. A179107, A179108, A179109, A179387, A179388
%K more,nonn,hard
%O 1,1
%A _Artur Jasinski_, Jul 12 2010, Jul 13 2010, Aug 03 2010
%E Edited by _M. F. Hasler_, Sep 30 2013
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