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A179784
Records for minima of the positive distance d between the seventh power of a positive integer x and the square of an integer y such that d = x^7 - y^2 (x <> k^2 and y <> k^7).
15
7, 71, 95, 448, 1756, 2215, 3983, 6271, 15231, 26775, 26870, 57475, 102703, 221916, 257963, 9053750, 9297464, 9321703, 27188154, 48787589, 62396287, 83146412, 152244535, 44475611670, 74378479183, 179884971502, 929051699593
OFFSET
1,1
COMMENTS
Distance d is equal to 0 when x = k^2 and y = k^7.
For x values see A179785.
For y values see A179786.
Conjecture (Artur Jasinski): For any positive number x >= A179785(n), the distance d between the seventh power of x and the square of any y (such that x <> k^2 and y <> k^7) can't be less than A179784(n).
MATHEMATICA
d = 7; 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