cp's OEIS Frontend

A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

%I A179388 #13 Jul 10 2015 00:42:31
%S A179388 5,11,181,207,225,500,524,1586,13537,376601,223063347,911054064,
%T A179388 16073515093,22143115844,29448160810,1661699554612,2498973838515,
%U A179388 26588790747913,27582731314539,178638660622364
%N A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.
%C A179388 "Records of minima" means values A179386(n)=A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently A181138(y) such that A181138(y') > A181138(y) for all y' > y. See the main entry A179386 for all further considerations. - _M. F. Hasler_, Sep 30 2013
%C A179388 For d values see A179386, for x values see A179387.
%C A179388 Theorem (_Artur Jasinski_):
%C A179388 For any positive number x >= A179387(n), the distance between the cube of x and the square of any y (with x<>n^2 and y<>n^3) can't be less than A179386(n).
%C A179388 Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable there can't exist any such x (or the related y).
%F A179388 A179388(n) = sqrt(A179387(n)^3 - A179386(n)).
%t A179388 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}]; yy (*Artur Jasinski*)
%Y A179388 Cf. A179107, A179108, A179109, A179387, A179388.
%Y A179388 Cf. A181138, A229618, A077116, A106265, A165288.
%K A179388 more,nonn,hard
%O A179388 1,1
%A A179388 _Artur Jasinski_, Jul 12 2010, Jul 13 2010, Aug 03 2010
%E A179388 Edited by _M. F. Hasler_, Sep 30 2013