OFFSET
1,2
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
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 *)
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Artur Jasinski, Jun 29 2010
STATUS
approved