4-Digit Numbers forming Longest Reverse-and-Add Sequence

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Theorem

Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.

Let $r \left({m}\right)$ be the reverse-and-add process on $m$.

Let $r$ be applied iteratively to $m$.


The $4$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are:

$6999, 7998, 8997, 9996$

all of which need $20$ iterations.


Proof

\((1):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 6999 + 9996 \, \) \(\, \displaystyle =\, \) \(\displaystyle 16995\) $\quad$ $\quad$
\((2):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 16995 + 59961 \, \) \(\, \displaystyle =\, \) \(\displaystyle 76956\) $\quad$ $\quad$
\((3):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 76956 + 65967 \, \) \(\, \displaystyle =\, \) \(\displaystyle 142923\) $\quad$ $\quad$
\((4):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 142923 + 329241 \, \) \(\, \displaystyle =\, \) \(\displaystyle 472164\) $\quad$ $\quad$
\((5):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 472164 + 461274 \, \) \(\, \displaystyle =\, \) \(\displaystyle 933438\) $\quad$ $\quad$
\((6):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 933438 + 834339 \, \) \(\, \displaystyle =\, \) \(\displaystyle 1767777\) $\quad$ $\quad$
\((7):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 1767777 + 7777671 \, \) \(\, \displaystyle =\, \) \(\displaystyle 9545448\) $\quad$ $\quad$
\((8):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 9545448 + 8445459 \, \) \(\, \displaystyle =\, \) \(\displaystyle 17990907\) $\quad$ $\quad$
\((9):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 17990907 + 70909971 \, \) \(\, \displaystyle =\, \) \(\displaystyle 88900878\) $\quad$ $\quad$
\((10):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 88900878 + 87800988 \, \) \(\, \displaystyle =\, \) \(\displaystyle 176701866\) $\quad$ $\quad$
\((11):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 176701866 + 668107671 \, \) \(\, \displaystyle =\, \) \(\displaystyle 844809537\) $\quad$ $\quad$
\((12):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 844809537 + 735908448 \, \) \(\, \displaystyle =\, \) \(\displaystyle 1580717985\) $\quad$ $\quad$
\((13):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 1580717985 + 5897170851 \, \) \(\, \displaystyle =\, \) \(\displaystyle 7477888836\) $\quad$ $\quad$
\((14):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 7477888836 + 6388887747 \, \) \(\, \displaystyle =\, \) \(\displaystyle 13866776583\) $\quad$ $\quad$
\((15):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 13866776583 + 38567766831 \, \) \(\, \displaystyle =\, \) \(\displaystyle 52434543414\) $\quad$ $\quad$
\((16):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 52434543414 + 41434543425 \, \) \(\, \displaystyle =\, \) \(\displaystyle 93869086839\) $\quad$ $\quad$
\((17):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 93869086839 + 93868096839 \, \) \(\, \displaystyle =\, \) \(\displaystyle 187737183678\) $\quad$ $\quad$
\((18):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 187737183678 + 876381737781 \, \) \(\, \displaystyle =\, \) \(\displaystyle 1064118921459\) $\quad$ $\quad$
\((19):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 1064118921459 + 9541298114601 \, \) \(\, \displaystyle =\, \) \(\displaystyle 10605417036060\) $\quad$ $\quad$
\((20):\quad\) \(\displaystyle \) \(\) \(\, \displaystyle 10605417036060 + 06063071450601 \, \) \(\, \displaystyle =\, \) \(\displaystyle 16668488486661\) $\quad$ $\quad$

which is palindromic.


$7998$ and its reversal converge on the same sequence immediately:

\(\displaystyle 7998 + 9996\) \(=\) \(\displaystyle 16995\) $\quad$ $\quad$



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