Kaprekar's Process on 5 Digit Number
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Theorem
Let $n$ be a $5$-digit integer whose digits are not all the same.
Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles:
- $53 \, 955 \to 59 \, 994 \to 53 \, 955$
- $61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$
- $62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964$
Proof
We have:
\(\ds 95 \, 553 - 35 \, 559\) | \(=\) | \(\ds 59 \, 994\) | ||||||||||||
\(\ds 99 \, 954 - 45 \, 999\) | \(=\) | \(\ds 53 \, 995\) |
\(\ds 97 \, 641 - 14 \, 679\) | \(=\) | \(\ds 82 \, 962\) | ||||||||||||
\(\ds 98 \, 622 - 22 \, 689\) | \(=\) | \(\ds 75 \, 933\) | ||||||||||||
\(\ds 97 \, 533 - 33 \, 579\) | \(=\) | \(\ds 63 \, 954\) | ||||||||||||
\(\ds 96 \, 543 - 34 \, 569\) | \(=\) | \(\ds 61 \, 974\) |
\(\ds 96 \, 642 - 24 \, 669\) | \(=\) | \(\ds 71 \, 973\) | ||||||||||||
\(\ds 97 \, 731 - 13 \, 779\) | \(=\) | \(\ds 83 \, 952\) | ||||||||||||
\(\ds 98 \, 532 - 23 \, 589\) | \(=\) | \(\ds 74 \, 943\) | ||||||||||||
\(\ds 97 \, 443 - 34 \, 479\) | \(=\) | \(\ds 62 \, 964\) |
This theorem requires a proof. In particular: It remains to be shown that all $5$-digit numbers end up here You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1972: Boris A. Kordemsky: The Moscow Puzzles: 359 Mathematical Recreations: $\text {XIII}$: Numbers Curious and Serious: $350$. A Persistent Difference
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $99,954$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $99,954$