Editing Pythagorean triple (section)

==Distribution of triples==
[[Image:Pythagorean triple scatterplot.svg|thumb|300px|A [[scatter plot]] of the legs {{math|(''a'',''b'')}} of the first Pythagorean triples with {{math|''a''}} and {{math|''b''}} less than 4500.]]

There are a number of results on the distribution of Pythagorean triples. In the scatter plot, a number of obvious patterns are already apparent. Whenever the legs {{math|(''a'',''b'')}} of a primitive triple appear in the plot, all integer multiples of {{math|(''a'',''b'')}} must also appear in the plot, and this property produces the appearance of lines radiating from the origin in the diagram.

Within the scatter, there are sets of [[Parabola|parabolic]] patterns with a high density of points and all their foci at the origin, opening up in all four directions. Different parabolas intersect at the axes and appear to reflect off the axis with an incidence angle of 45 degrees, with a third parabola entering in a perpendicular fashion. Within this quadrant, each arc centered on the origin shows that section of the parabola that lies between its tip and its intersection with its [[semi-latus rectum]].

These patterns can be explained as follows. If <math>a^2/4n</math> is an integer, then ({{math|''a''}}, <math>|n-a^2/4n|</math>, <math>n+a^2/4n</math>) is a Pythagorean triple. (In fact every Pythagorean triple {{math|(''a'', ''b'', ''c'')}} can be written in this way with integer {{math|''n''}}, possibly after exchanging {{math|''a''}} and {{math|''b''}}, since <math>n=(b+c)/2</math> and {{math|''a''}} and {{math|''b''}} cannot both be odd.) The Pythagorean triples thus lie on curves given by <math>b = |n-a^2/4n|</math>, that is, parabolas reflected at the {{math|''a''}}-axis, and the corresponding curves with {{math|''a''}} and {{math|''b''}} interchanged. If {{math|''a''}} is varied for a given {{math|''n''}} (i.e. on a given parabola), integer values of {{math|''b''}} occur relatively frequently if {{math|''n''}} is a square or a small multiple of a square. If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip. For instance, {{math|1=38{{sup|2}} = 1444}}, {{math|1=2 × 27{{sup|2}} = 1458}},
{{math|1=3 × 22{{sup|2}} = 1452}}, {{math|1=5 × 17{{sup|2}} = 1445}} and {{math|1=10 × 12{{sup|2}} = 1440}}; the corresponding parabolic strip around {{math|1=''n'' ≈ 1450}} is clearly visible in the scatter plot.

The angular properties described above follow immediately from the functional form of the parabolas. The parabolas are reflected at the {{math|''a''}}-axis at {{math|1=''a'' = 2''n''}}, and the derivative of {{math|''b''}} with respect to {{math|''a''}} at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value {{math|2''n''}} also corresponds to a cluster. The corresponding parabola intersects the {{math|''b''}}-axis at right angles at {{math|1=''b'' = 2''n''}}, and hence its reflection upon interchange of {{math|''a''}} and {{math|''b''}} intersects the {{math|''a''}}-axis at right angles at {{math|1=''a'' = 2''n''}}, precisely where the parabola for {{math|''n''}} is reflected at the {{math|''a''}}-axis. (The same is of course true for {{math|''a''}} and {{math|''b''}} interchanged.)

Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.<ref>[http://conservancy.umn.edu/handle/4878 1988 Preprint] {{Webarchive|url=https://web.archive.org/web/20110809123211/http://conservancy.umn.edu/handle/4878 |date=2011-08-09 }} See Figure 2 on page 3., later published as {{citation |first=Albert |last=Fässler |title=Multiple Pythagorean number triples |journal=American Mathematical Monthly |volume=98 |issue=6 |pages=505–517 |date=June–July 1991 |jstor=2324870 |doi=10.2307/2324870|url=http://purl.umn.edu/4878 }}</ref><ref>{{citation |first1=Manuel |last1=Benito |first2=Juan L. |last2=Varona |title=Pythagorean triangles with legs less than {{math|''n''}} |journal=Journal of Computational and Applied Mathematics |volume=143 |pages=117–126 |date=June 2002 |issue=1 |doi=10.1016/S0377-0427(01)00496-4|bibcode=2002JCoAM.143..117B |doi-access=free }} as [http://www.unirioja.es/cu/jvarona/downloads/Benito-Varona-JCAM-Publicado.pdf PDF]</ref>