How do I visualization generalized Pythagorean theory?

Question

maybe the Pythagorean equation is not alone. can't this equation be just one element of an array like this for example. exponents must be prime numbers (otherwise it's just a combination of other elements)

first elment a^2+b^2=c^2 (this is Pythagoras) second element a^3+b^3+c^3=d^3 ( but I can't visualize it in the coordinate system like Pythagoras equation) thirth element a^5+b^5+c^5+d^5+e^5=f^5 fourth element a^7+b^7+c^7...et

Answer 1

Clear["Global`*"]

To bound the problem let 1 <= {a, b, c} <= 100 and to avoid equivalent results that are just reorderings of {a, b, c} let {a, b, c} be ordered such that a <= b <= c <= 100,

(inst = Select[
    Flatten[
     Table[{a, b, c, Norm[{a, b, c}, 3]},
      {a, 1, 100}, {b, a, 100}, {c, b, 100}],
     2],
    IntegerQ@Last@# &]) // Length

(* 98 *)

Verifying,

And @@ (Total[Most[#]^3] == Last[#]^3 & /@ inst)

(* True *)

Looking at the first five and last five instances,

TableForm[inst[[{1, 2, 3, 4, 5, 94, 95, 96, 97, 98}]], 
 TableHeadings -> {None, {a, b, c, d}},
 TableAlignments -> {Right, Center}]

{dmin, dmax} = MinMax[inst[[All, 4]]]

(* {6, 139} *)

Plotting,

Legended[
 ListPointPlot3D[Most /@ inst,
  AxesLabel -> (Style[#, 14] & /@ {a, b, c}),
  ColorFunction -> Function[{a, b, c},
    ColorData["Rainbow"][
     Rescale[Norm[{a, b, c}, 3],
      {dmin, dmax}]]],
  ColorFunctionScaling -> False],
 BarLegend[{"Rainbow", {dmin, dmax}},
  LegendLabel -> Style[d, 14]]]