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Revision difference : Pythagorean triplet |
| Version 9 |
Version 8 |
| A \emph{Pythagorean triplet} is a set $\{a, b, c\}$ of three positive |
A \emph{Pythagorean triplet} is a set $\{a, b, c\}$ of three positive |
| integers such that |
integers such that |
| \[ |
\[ |
| a^2 + b^2 = c^2. |
a^2 + b^2 = c^2. |
| \] |
\] |
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| That is, $\{a, b, c\}$ is a Pythagorean triplet if there exists a |
That is, $\{a, b, c\}$ is a Pythagorean triplet if there exists a |
| right triangle whose sides have lengths $a$, $b$, and $c$, |
right triangle whose sides have lengths $a$, $b$, and $c$, |
| respectively. For example, $\{3, 4, 5\}$ is a Pythagorean triplet. |
respectively. For example, $\{3, 4, 5\}$ is a Pythagorean triplet. |
| Given one Pythagorean triplet $\{a, b, c\}$, we can produce another by |
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 |
multiplying $a$, $b$, and $c$ by the same factor $k$. It follows that |
| there are countably many Pythagorean triplets. |
there are countably many Pythagorean triplets. |
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| \subsubsection*{Primitive Pythagorean triplets} |
\subsubsection*{Primitive Pythagorean triplets} |
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| A Pythagorean triplet is \emph{primitive} if its elements are |
A Pythagorean triplet is \emph{primitive} if its elements are |
| coprimes. All primitive Pythagorean triplets are given by |
coprimes. All primitive Pythagorean triplets are given by |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| a &=& 2mn,\\ |
a &=& 2mn,\\ |
| b &=& m^2\!-\!n^2,\\ |
b &=& m^2\!-\!n^2,\\ |
| c &=& m^2\!+\!n^2, |
c &=& m^2\!+\!n^2, |
| \end{eqnarray*} |
\end{eqnarray*} |
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| where the \emph{seed numbers} $m$ and $n$ are any two coprime positive |
where the \emph{seed numbers} $m$ and $n$ are any two coprime positive |
| integers, one odd and one even, such that $m > n$. |
integers, one odd and one even, such that $m > n$. |
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\textbf{Note.}\, One can form the sequence (cf. Sloane's \PMlinkexternal{A100686}{http://www.research.att.com/~njas/sequences/?q=A100686&language=english&go=Search})
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\textbf{Note.}\, One can form the sequence (Sloane's A100686)
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| \[ |
\[ |
| 1,\,2,\,3,\,4,\,7,\,24,\,527,\,336,\,164833,\,354144,\,... |
1,\,2,\,3,\,4,\,7,\,24,\,527,\,336,\,164833,\,354144,\,... |
| \] |
\] |
| taking first the seed numbers 1 and 2 which give the legs 3 and 4, |
taking 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 |
taking these as new seed numbers which give the legs 7 and 24, and |
| so on. |
so on. |
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| % related: PythagoreanTriplesAndRationalPointsOnAUnitHyperbola |
% related: PythagoreanTriplesAndRationalPointsOnAUnitHyperbola |
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