A179784 - OEIS

A179784

Records for minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

%I #10 Sep 08 2023 22:40:00

%S 7,71,95,448,1756,2215,3983,6271,15231,26775,26870,57475,102703,

%T 221916,257963,9053750,9297464,9321703,27188154,48787589,62396287,

%U 83146412,152244535,44475611670,74378479183,179884971502,929051699593

%N Records for minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).

%C Distance d is equal to 0 when x = k^2 and y = k^7.

%C For x values see A179785.

%C For y values see A179786.

%C Conjecture (_Artur Jasinski_): For any positive number x >= A179785(n), the distance d between the seventh power of x and the square of any y (such that x <> k^2 and y <> k^7) can't be less than A179784(n).

%t d = 7; 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, A179785, A179786.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jul 27 2010