A179107 Records of minima of positive distance d between a square cubefree integer y and a cube of positive and squarefree integer x and such d = y^2 - x^3.

1, 8, 15, 17, 24, 225, 1090, 8569, 11492, 14668, 14857, 28024, 117073

Offset: 1

Author

Artur Jasinski, Jun 29 2010

Keywords

• more
• nonn

Comments

If x=n^2 and y=n^3 distance d=0.

For x values see A179108.

For y values see A179109.

For numbers x from 46 to 108 distance can't be less than 8.

For numbers x from 109 to 5233 distance can't be less than 15.

For numbers x from 5234 to 8157 distance can't be less than 17.

For numbers x from 8158 to 729113 distance can't be less than 24.

For numbers x from 729114 to 28187350 distance can't be less than 225.

Next conjectured terms are: 4401169, 87002345, 193234265, 497218657.

Links

• J. Calvo, J. Herranz, G. Saez, A new algorithm to search for small nonzero |x^3 - y^2| values, Math. Comp. 78 (2009), 2435-2444.
• N. Elkies, Hall conjecture

Crossrefs

Cf. A179108, A179109, A179386, A179387, A179388.

Programs

Mathematica

d = 3; 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)] + 1; k = m^2 - n^d; 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, 720114}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd (* Artur Jasinski, Oct 30 2011 *)