Pythagorean tripletPythagoreanTriple2001-10-06 17:19:422008-12-05 11:29:11Definitionseed numberprimitive Pythagorean tripleprimitive Pythagorean tripletTrianglePythagorasGeometry\usepackage{amssymb}
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\usepackage{xypic}A \emph{Pythagorean triplet} is a set $\{a, b, c\}$ of three positive
integers such that
\[
a^2 + b^2 = c^2.
\]
That is, $\{a, b, c\}$ is a Pythagorean triplet if there exists a
right triangle whose sides have lengths $a$, $b$, and $c$,
respectively. For example, $\{3, 4, 5\}$ is a Pythagorean triplet.
Given one Pythagorean triplet $\{a, b, c\}$, we can produce another by
multiplying $a$, $b$, and $c$ by the same factor $k$. It follows that
there are countably many Pythagorean triplets.
\subsubsection*{Primitive Pythagorean triplets}
A Pythagorean triplet is \emph{primitive} if its elements are
coprimes. All primitive Pythagorean triplets are given by
\begin{align}
\begin{cases}
a = 2mn,\\
b = m^2\!-\!n^2,\\
c = m^2\!+\!n^2,
\end{cases}
\end{align}
where the \emph{seed numbers} $m$ and $n$ are any two coprime positive
integers, one odd and one even, such tht $m > n$.\, The equations (1) give all Pythagorean triplets, if one presumes of the positive integers $m$ and $n$ only that\, $m > n$.\\
\textbf{Note 1.}\, Among the primitive Pythagorean triples, the odd cathetus $a$ may attain all odd values except 1 (set e.g.\, $m := n\!+\!1$) and the even cathetus $b$ all values divisible by 4 (set\, $n := 1$).\\
\textbf{Note 2.}\, In the primitive triples, the hypothenuses $c$ are always odd.\, All possible Pythagorean hypotenuses are contraharmonic means of two different integers (and conversely).\\
\textbf{Note 3.}\, One can form the sequence (cf. Sloane's \PMlinkexternal{A100686}{http://www.research.att.com/~njas/sequences/?q=A100686&language=english&go=Search})
\[
1,\,2,\,3,\,4,\,7,\,24,\,527,\,336,\,164833,\,354144,\,...
\]
taking first the seed numbers 1 and 2 which give the legs 3 and 4,
taking these as new seed numbers which give the legs 7 and 24, and
so on.
% related: PythagoreanTriplesAndRationalPointsOnAUnitHyperbola