A179786 - OEIS

A179786

Values y for records of the 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 22:41:56

%S 11,46,529,1448,3162,10267,24743,35777,116159,147885,370447,1233870,

%T 1577546,6392774,211053546,264783325,272123427,1011697339,1140219273,

%U 2370360092,49411058753,63606986977,71996746561757,137783827309893

%N Values y for records of the 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 d values see A179784.

%C For x values see A179785.

%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}]; yy

%Y Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408, A179784, A179785.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jul 27 2010