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A179790
Records for minima of the positive distance d between the ninth power of a positive integer x and the square of an integer y such that d = x^9 - y^2 (x <> k^2 and y <> k^9).
12
28, 83, 1516, 3420, 5503, 30889, 75228, 776563, 2428283, 3035356, 29901479, 68334642, 113284785, 776887258, 1719856432, 3353407292, 19232010711, 27678166236, 29160146546, 305337557432, 95950163566107, 114852386371373
OFFSET
1,1
COMMENTS
Distance d is equal to 0 when x = k^2 and y = k^9.
For x values see A179791.
For y values see A179792.
Conjecture (Artur Jasinski): For any positive number x >= A179791(n), the distance d between the ninth power of x and the square of any y (such that x <> k^2 and y <> k^9) can't be less than A179790(n).
MATHEMATICA
d = 9; 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
KEYWORD
nonn
AUTHOR
Artur Jasinski, Jul 27 2010
STATUS
approved