As an addendum, here is a quick way to get a whole pile of triples, using PowersRepresentations[] (the same function used by the OP in the function sameValuePPTTable[]):
Column[Row[#, Spacer[5]] & /@
Apply[Row[{#1^Defer[2], "+", #2^Defer[2], "=", Sqrt[#1^2 + #2^2]^Defer[2]}] &,
DeleteCases[Table[DeleteCases[PowersRepresentations[n^2, 2, 2], {0, _}],
{n, 5, 101}], {}], {2}]]
