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%I #17 Sep 08 2023 22:39:20
%S 7,7538,283261,494576,4235622,7135951,38053824,55905695,185380312,
%T 1208691743,3263221507,14034746735,14732727599,24211719874,
%U 68491624661,136264246246,5337970328375,6845918569200,15505738619231,30037885135088
%N Record minima of the positive distance d between the fifteenth power of a positive integer x and the square of an integer y such that d = x^15 - y^2 (x <> k^2 and y <> k^15).
%C Distance d is equal to 0 when x = k^2 and y = k^15.
%C For x values see A179813.
%C For y values see A179814.
%C Conjecture: For any positive number x >= A179813(n), the distance d between the fifteenth power of x and the square of any y (such that x <> k^2 and y <> k^15) can't be less than A179812(n).
%t d = 15; max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^d)^(1/2)]; k = n^d - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 10000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd
%Y Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795, A179798, A179799, A179800, A179812, A179813, A179814.
%K nonn
%O 1,1
%A _Artur Jasinski_, Jul 28 2010
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