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A179792
Values y for records of the 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
22, 140, 1397, 3174, 11585, 102978, 1098758, 1342070, 2761448, 116348986, 326908123, 5661454305, 14439547606, 24195364585, 44988513611, 1037782490126, 18907836782131, 50577039498042, 476237361126871, 10815891488601655
OFFSET
1,1
COMMENTS
Distance d is equal to 0 when x = k^2 and y = k^9.
For d values see A179790.
For x values see A179791.
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}]; yy
KEYWORD
nonn
AUTHOR
Artur Jasinski, Jul 27 2010
STATUS
approved