Author James M Parks June 14, 2021

The graph of a Pythagorean triple (a,b,c) is the graph of the pair (a,b) in the xy-plane. We are interested in the graph of primitive Pythagorean triples (PPTs), GDC(a,b,c)=1.
Calculations use a generalization of the Pythagorean/Plato method, and the difference triple form (a,b,b+d), where the difference d belongs to the sequence OEIS:A096033, thus b(a) = (a^2-d^2)/2d, and this determines a parabola in the graph of the PPTs. The Data Tables of PPTs for each d value k in OEIS:A096033 use the code below for dk, k=1,...,625.
The graph of {dk, k=1, 2,...} is ListPlot s1. The reflected data sets, rt[dk], are dk reflected about the line y = x, and satisfy a > b, see ListPlot s2. The graph of the combined Plots is Show[{s1,s2}], and the parabolas are visible in this graph.
Compare this result with the graph by R. Knott, Pythagorean Right Triangles, http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#section3.4 , which uses a different Mathematica program.

In[]:=

ColorOutputRGBColor

rt=ReflectionTransform[{1,-1}]

dk=Table[With[{max=1720},Map[Last,List@@(Reduce[x^2==2ky+k^2&&0<x<y<max&&GCD[x,y]==1,{x,y},Integers,BacksubstitutionTrue]/.AndList),{2}]]]rt[dk]

s1=ListPlot[{d1,d2,d8,d9,d18,d25,d32,d49,d50,d72,d81,d98,d121,d128,d162,d169,d200,d225,d242,d288,d289,d338,d361,d392,d441,d450,d512,d529,d578,d625},PlotStyle{RGBColor[1,0,0]},AspectRatioAutomatic]

s2=ListPlot[{rt[d1],rt[d2],rt[d8],rt[d9],rt[d18],rt[d25],rt[d32],rt[d49],rt[d50],rt[d72],rt[d81],rt[d98],rt[d121],rt[d128],rt[d162],rt[d169],rt[d200],rt[d225],rt[d242],rt[d288],rt[d289],rt[d338],rt[d361],rt[d392],rt[d441],rt[d450],rt[d512],rt[d529],rt[d578],rt[d625]},PlotStyle{RGBColor[0,0.1,0.1]},AspectRatioAutomatic]

Show[{s1,s2},PlotRangeAll]

Cite this as: James M Parks, "PPT GraphBrief.nb" from the Notebook Archive (2021), https://notebookarchive.org/2021-06-6y24ve8