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Revision difference : Pythagorean triplet |
| Version 6 |
Version 5 |
| A \emph{Pythagorean triplet} is a set\, $\{a,\,b,\,c\}$\, of three integers such that |
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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$$ |
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| That is, $\{a,\,b,\,c\}$ is a Pythagorean triplet if there exists a right triangle whose sides are $a,\,b,\,c$.\, An example is\, $\{3,\,4,\,5\}$.\, |
That is, $\{a,\,b,\,c\}$ is a Pythagorean triplet if there exists a right triangle whose sides are $a,\,b,\,c$.\, An example is\, $\{3,\,4,\,5\}$.\, |
| If\, $\{a,\,b,\,c\}$\, a Pythagorean triplet, so is\, $\{ka,\,kb,\,kc\}$\, for\, $k = 1,\,2,\,\ldots$.\, It follows that there are countably many Pythagorean triplets. |
If\, $\{a,\,b,\,c\}$\, a Pythagorean triplet, so is\, $\{ka,\,kb,\,kc\}$\, for\, $k = 1,\,2,\,\ldots$. |
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It follows that there are countably many Pythagorean triplets. |
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| \subsubsection*{Primitive Pythagorean triplets} |
\subsubsection*{Primitive Pythagorean triplets} |
| If $a,\,b,\,c$ are coprimes, then we say that the triplet is \emph{primitiv{e}}.\, All the primitive Pythagorean triplets are given by |
If $a,\,b,\,c$ are coprimes, then we say that the triplet is \emph{primitiv{e}}.\, |
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All the primitive Pythagorean triplets are given by |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| a &=& 2mn,\\ |
a &=& 2mn,\\ |
| b &=& m^2\!-\!n^2,\\ |
b &=& m^2\!-\!n^2,\\ |
| c &=& m^2\!+\!n^2, |
c &=& m^2\!+\!n^2, |
| \end{eqnarray*} |
\end{eqnarray*} |
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where the {\em seed numbers} $m,\,n$ are any two coprime positive integers, one odd and the other even with\, $m > n$.
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where the {\em seed numbers} $m,\,n$ are any two coprime integers, one odd and the other even with $m > n$.
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| \textbf{Note.}\, One can form the sequence (Sloane's A100686) |
\textbf{Note.}\, One can form the sequence (Sloane's A100686) |
| $$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 cathets 3 and 4, taking these as new seed numbers which give the cathets 7 and 24, and so on. |
taking first the seed numbers 1 and 2 which give the cathets 3 and 4, taking these as new seed numbers which give the cathets 7 and 24, and so on. |
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