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Revision difference : Pythagorean triplet |
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Version 3 |
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A \emph{Pythagorean triplet} is a set\, $\{a,\,b,\,c\}$\, of three integers such that
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A \emph{Pythagorean triplet} is a set $\{a,b,c\}$ of three integers such that
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| a^2+b^2=c^2. |
a^2+b^2=c^2. |
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$$ |
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That is, $\{a,\,b,\,c\}$ is a Pythagorean triplet if there exists a right triangle whose sides are $a,\,b,\,c$.\, An example is\, $\{3,\,4,\,5\}$.\,
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That is, $\{a,b,c\}$ is a Pythagorean triplet if there exists a right triangle whose sides are $a,b,c$. An example is $\{3,4,5\}$.
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If\, $\{a,\,b,\,c\}$\, a Pythagorean triplet, so is\, $\{ka,\,kb,\,kc\}$\, for\, $k = 1,\,2,\,\ldots$.
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If $\{a,b,c\}$ a Pythagorean triplet, so is $\{ka,kb,kc\}$ for $k=1,2,\ldots$.
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| It follows that there are countably many Pythagorean triplets. |
It follows that there are countably many Pythagorean triplets. |
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| \subsubsection*{Primitive Pythagorean triplets} |
\subsubsection*{Primitive Pythagorean triplets} |
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If $a,\,b,\,c$ are coprimes, then we say that the triplet is \emph{primitiv{e}}.\,
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If $a,b,c$ are coprimes, then we say that the triplet is \emph{primitiv{e}}.
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| All the primitive Pythagorean triplets are given by |
All the primitive Pythagorean triplets are given by |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| a &=& 2mn,\\ |
a &=& 2mn,\\ |
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b &=& m^2\!-\!n^2,\\
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b &=& m^2-n^2,\\
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c &=& m^2\!+\!n^2,
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c &=& m^2+n^2,
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| \end{eqnarray*} |
\end{eqnarray*} |
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where the {\em seed numbers} $m,\,n$ are any two coprime integers, one odd and the other even with $m > n$.
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where $m,n$ are any two coprime integers, one odd and the other even with $m>n$.
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