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A179791
Values x 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
2, 3, 5, 6, 8, 13, 22, 23, 27, 62, 78, 147, 181, 203, 233, 468, 892, 1110, 1827, 3657, 3723, 10637, 11145, 11478, 12275, 16764, 19151, 22719, 23580, 24974, 30163, 36885, 41759, 41948, 44427, 66443, 86167, 96658, 115992, 222962, 248461, 248588, 384573
OFFSET
1,1
COMMENTS
Distance d is equal to 0 when x = k^2 and y = k^9.
For d values see A179790.
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}]; xx
KEYWORD
nonn
AUTHOR
Artur Jasinski, Jul 27 2010
STATUS
approved