Benedict Irwin edited The 21 thing.tex  over 9 years ago

Commit id: 5d518cb8b3804680a3d21d03f0c2df4b8d31a977

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Find $(21;;n)$ primality, \begin{equation}  \begin{array}{|c|c|c|}  \hline  n & prime? & fact \\ & rewrite\\  \hline  (21;;1) & 0 & 3×7 & (21;;1)  \\ (21;;2) & 0 & 3×7×101 \\ & (21;;1)×101\\  (21;;3) & 0 & 3^2×7^2×13×37 \\ &(21;;1)^2×13×37\\  (21;;4) & 0 & 3×7×73×101×137 \\ &(21;;2)×73×137\\  (21;;5) & 0 & 3×7×41×271×9091 \\ &\\  (21;;6) & 0 & 3^2×7^2×13×37×101×9901 \\ & \frac{(21;;2)×(21;;3)×9901}{(21;;1)}\\  (21;;7) & 0 & 3×7×239×4649×909091 \\  (21;;8) & 0 & 3×7×17×73×101×137×5882353 \\  (21;;9) & 0 & 3^3×7^2×13×19×37×52579×333667 \\ 

\end{array}  \end{equation}  Each sequence seems individual and fascinating. Everything seems to have a factor of $3$,$7$. However our old friends $9091$ and $9901$ appear very soon into the sequence at $n=5,6$, funnily enough begin $10,12$ divided by two, which were the $11...$ sequences positions. Also $5882353$ makes an appearence at $8$ which is $16$ by two, $16$ begin the $111...$ sequence counterpart. Equally our friend \begin{equation}  9901 = \frac{(21;;1)(21;;6)}{(21;;2)(21;;3)}  \end{equation}