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%I #10 Sep 08 2023 22:39:32
%S 2,3,6,7,8,10,14,18,20,26,28,32,38,52,60,77,145,168,222,237,268,279,
%T 286,359,367,390,536,569,622,872,1085,1349,1462,1760,1932,2423,2801,
%U 5559,5925,7052,8383,8752,10075,11917,15712,17420,17598,23712,26026,28095
%N Values x for records of the minima of the positive distance d between the eleventh power of a positive integer x and the square of an integer y such that d = x^11 - y^2 (x <> k^2 and y <> k^11).
%C Distance d is equal to 0 when x = k^2 and y = k^11.
%C For x values see A179794.
%C For x values see A179795.
%C Conjecture (_Artur Jasinski_): For any positive number x >= A179794(n), the distance d between the eleventh power of x and the square of any y (such that x <> k^2 and y <> k^11) can't be less than A179793(n).
%t d = 11; 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}]; xx
%Y Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.
%K nonn
%O 1,1
%A _Artur Jasinski_, Jul 27 2010
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