Talk:Kaprekar number

Following the description on this page, how can 1 be a Kaprekar number? One squared is one, which cannot be split in to two one-digit numbers.

Therefore, either one is not a Kaprekar number, or the description on this page needs a special case for the number one. --Dan Huby 14:55, 6 Apr 2004 (UTC)

According to http://www.math.uwaterloo.ca/JIS/VOL3/iann2a.html, the two split digits don't have to be the same size, and one can be zero-length, if I've read it correctly. Which gives a case for one. But then the text on this page needs extending. --Dan Huby 15:11, 6 Apr 2004 (UTC)

Usefulness
What is the usefulness of a Kaprekar number? - Zepheus (ツィフィアス) 17:17, 3 August 2006 (UTC) Yes. Speaking as a layman, "why is this a thing"? --Richardson mcphillips (talk) 02:56, 9 March 2015 (UTC)

Odd number of digits
Since the two parts don't need to be of equal length, a Kaprekar number can have an odd number of digits. 2607:FB90:2113:EA80:B5CB:1FF3:4674:FBF5 (talk) 20:06, 31 May 2015 (UTC)

Corrections
Given that no sources have been provided for any of the lists in the Other bases section of the article, I have written a script for a symbolic algebra package to check them. The output of this script agrees with the article's lists for decimal and hexadecimal, but uncovered problems with the duodecimal and base 7 lists.

My script confirms that all the numbers given in the list of duodecimal Kaprekar numbers are indeed Kaprekar, but found two more, 4873012 and 7239212, which were missing from the list. That these are also Kaprekar is easy to check with a calculator:
 * 48730122 = 1X3004690012  and   4873012 = 04690012 + 1X3012
 * 72392122 = 438X06X00412  and   7239212 = 06X00412 + 438X12.

There were many more problems with the base 7 list. According to my script, more than two thirds of the numbers given in this list are not Kaprekar, and there are numerous Kaprekar numbers missing from it. I have not checked any more than a handful of these with an explicit hand calculation, but for all those I have checked, the calculation has vindicated the output of my script. One example is the number 347, which had been given in the article's list. But  3472 = 15527. There are only 3 ways to split this latter number into two parts and add them, 1557 + 27, 157 + 527  and  17 + 5527, and none of these sums is equal to 347. Therefore 347 cannot be Kaprekar.

On the other hand, 227, which was not given in the list of base 7 Kaprekar numbers, is easily verified to be Kaprekar, since
 * 2272 = 5147, and  227 = 57 + 147

David Wilson (talk · cont) 11:22, 25 September 2017 (UTC) P.S. I see that the both the erroneous list of septimal Kaprekar numbers and the errors in the list of duodecimal Kaprekar numbers, as well as several other errors, now corrected, were introduced during of edits, from an IP that has also been used to make. David Wilson (talk · cont)

4879 and 5292 are Kaprekar
48792 = 23804641, and 4879 = 238 + 04641 52922 = 28005264, and 5292 = 28 + 005292 The decimal expansions of the numbers you add together are allowed to include leading zeroes, as long as they do not constitute the whole of the expansion. David Wilson (talk · cont) 22:56, 9 October 2017 (UTC) P.S. As is 38962. These numbers have been removed several times by editors who appear not to realise that (as the article itself points out) the addend taken from the second half of the square is allowed to contain leading zeroes. For ease of reference, here's a list of all those numbers from the article's list of decimal Kaprekar numbers for which this phenomenon occurs:
 * 992 = 9801 = (98 + 01)2.
 * 9992 = 998001 = (998 + 001)2.
 * 48792 = 23804641 = (238 + 04641)2
 * 52922 = 28005264 = (28 + 005292)2
 * 99992 = 99980001 = (9998 + 0001)2.
 * 389622 = 1518037444 = (1518 + 037444)2.
 * 951212 = 9048004641 = (90480 + 04641)2.
 * 999992 = 9999800001 = (99998 + 00001)2.

David Wilson (talk · cont) 00:28, 19 December 2017 (UTC)

0 a Kaprekar number?
This page seems to be the only one that accepts 0 and any powers of 10 (resp. any powers of base) as Kaprekar numbers. In any other page dealing with K numbers, these numbers are excluded. CKWG — Preceding unsigned comment added by 185.22.143.114 (talk) 15:12, 15 November 2019 (UTC)

Proof of finite p-Kaprekar numbers may be incorrect
Hello, I think the proof for finiteness of $$p$$-Kaprekar numbers may be incorrect (located under the Definition and properties section).

The proof claims that $$\beta = m^2$$ for the given example; however, if you compute $$\beta$$, it's actually

\begin{align} \beta & = n^2 \bmod b^p \\ & = ((b^p + 2m)b^p + m^2) \bmod b^p \\ & = m^2 \bmod b^p \\ \end{align} $$

Note that this is an important claim in the proof because it may be the case that $$m^2 \ge b^p $$. For instance, consider $$n=19, b = 10,$$ and $$ p =1$$. In this case, $$b^p = 10$$ and $$m = 9$$. Which means that $$m^2 > b^p$$ because $$81 > 10$$.

The proof is still structurally correct. Below is the fix I believe is necessary for the proof.

Along with the re-computing $$\beta$$, we need to recompute $$\alpha$$. If you let $$ \beta = m^2 \bmod b^p $$ then $$ \alpha = b^p + 2m + \frac{m^2-(m^2\bmod b^p)}{b^p}$$. Therefore,


 * $$ \alpha + \beta = n + m + \frac{m^2-(m^2\bmod b^p)}{b^p} + m^2\bmod b^p $$
 * $$ > n $$

Please let me know if I'm missing anything. Thank you for reviewing this!

KamyarGhiam (talk) 00:03, 10 August 2023 (UTC)