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Values x for records of minima of the positive distance d between an 11th power of a positive integer x and a square of an integer y such that d = x^13 - y^2 (x<>k^2 and y<>k^13).
6

%I #8 Jun 14 2021 17:51:36

%S 2,3,5,6,8,11,13,14,23,24,35,40,42,45,50,54,62,70,79,85,88,89,142,152,

%T 220,345,353,364,412,416,455,627,734,743,911,921,1068,1095,1294,1894,

%U 2398,2719,2887,3015,3623,3814,5837,6226,8603,8669,8971,9987,12683

%N Values x for records of minima of the positive distance d between an 11th power of a positive integer x and a square of an integer y such that d = x^13 - y^2 (x<>k^2 and y<>k^13).

%C Distance d = 0 when x = k^2 and y = k^13.

%C For d values see A179798.

%C For y values see A179800.

%C Conjecture: For any positive number x >= A179799(n) the distance d between the 11th power of x and the square of any y (such that x<>k^2 and y<>k^13) can't be less than A179798(n).

%t d = 13; 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, A179798, A179799, A179800.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jul 27 2010