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A179388
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Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.
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30
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5, 11, 181, 207, 225, 500, 524, 1586, 13537, 376601, 223063347, 911054064, 16073515093, 22143115844, 29448160810, 1661699554612, 2498973838515, 26588790747913, 27582731314539, 178638660622364
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OFFSET
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1,1
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COMMENTS
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For any positive number x >= A179387(n), the distance between the cube of x and the square of any y (with x<>n^2 and y<>n^3) can't be less than A179386(n).
Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable there can't exist any such x (or the related y).
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LINKS
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FORMULA
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MATHEMATICA
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max = 1000; vecd = Table[10100, {n, 1, max}]; vecx = Table[10100, {n, 1, max}]; vecy = Table[10100, {n, 1, max}]; len = 1; min = 10100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; yy (*Artur Jasinski*)
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CROSSREFS
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KEYWORD
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more,nonn,hard
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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