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

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

which is palindromic.


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

\(\displaystyle 7998 + 8997\) \(=\) \(\displaystyle 16995\)



Sources