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A179799
Values x for records of minima of the positive distance d between an 11th power of a positive integer x and a square of an integer y such that d = x^13 - y^2 (x<>k^2 and y<>k^13).
6
2, 3, 5, 6, 8, 11, 13, 14, 23, 24, 35, 40, 42, 45, 50, 54, 62, 70, 79, 85, 88, 89, 142, 152, 220, 345, 353, 364, 412, 416, 455, 627, 734, 743, 911, 921, 1068, 1095, 1294, 1894, 2398, 2719, 2887, 3015, 3623, 3814, 5837, 6226, 8603, 8669, 8971, 9987, 12683
OFFSET
1,1
COMMENTS
Distance d = 0 when x = k^2 and y = k^13.
For d values see A179798.
For y values see A179800.
Conjecture: For any positive number x >= A179799(n) the distance d between the 11th power of x and the square of any y (such that x<>k^2 and y<>k^13) can't be less than A179798(n).
MATHEMATICA
d = 13; 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