Pairs of numbers x > y > 0 such that x^3 + y^3 is a square, x is as small as possible, and x/y is unique.
2, 1, 2, 2, 21, 7, 26, 22, 37, 11, 46, 2, 65, 10, 65, 56, 70, 14, 88, 33, 91, 65, 104, 65, 105, 70, 105, 84, 112, 57, 190, 114, 249, 183, 260, 65, 273, 78, 312, 217, 330, 154, 345, 184, 345, 330, 354, 78, 364, 273, 371, 329, 407, 37, 433, 242, 450, 34, 456, 57
1
As the exponent is increased, it becomes harder to find terms.
T. D. Noe, Plot of 1000 pairs
T. D. Noe, Table of 1000 pairs
(Mma) mx = 1000; s3 = {}; t3 = {}; Do[If[IntegerQ[Sqrt[a^3 + b^3]] && ! MemberQ[s3, a/b], AppendTo[s3, a/b]; AppendTo[t3, {a, b}]], {a, 0, mx}, {b, a}]; t3
nonn
T. D. Noe, Apr 06 2017