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A179791 Values x for records of the minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).

%I A179791 #12 Sep 08 2023 22:42:08
%S A179791 2,3,5,6,8,13,22,23,27,62,78,147,181,203,233,468,892,1110,1827,3657,
%T A179791 3723,10637,11145,11478,12275,16764,19151,22719,23580,24974,30163,
%U A179791 36885,41759,41948,44427,66443,86167,96658,115992,222962,248461,248588,384573
%N A179791 Values x for records of the minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).
%C A179791 Distance d is equal to 0 when x = k^2 and y = k^9.
%C A179791 For d values see A179790.
%C A179791 For y values see A179792.
%C A179791 Conjecture (_Artur Jasinski_): For any positive number x >= A179791(n), the distance d between the ninth power of x and the square of any y (such that x <> k^2 and y <> k^9) can't be less than A179790(n).
%t A179791 d = 9; 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 A179791 Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785, A179786, A179790, A179791, A179792, A179793, A179794, A179795.
%K A179791 nonn
%O A179791 1,1
%A A179791 _Artur Jasinski_, Jul 27 2010