22222^2

[ivo zerkov]

Consider this: 22222^2=493817284

while 4938+17284=22222.

Is this well-known? I've attached a rough explanation for why it occurs. 

[Seiichi Manyama]

Dear ivo,

I have examined integers with the same properties in the past. Of course I found what you found.

https://manchanr6.blogspot.com/2019/10/191006.html

Seiichi

2025年3月28日金曜日 13:03:18 UTC+9 zerk...@gmail.com:

[Charles Kusniec]

Hi Ivo,

That's a very elegant observation. While I'm not aware of a named theorem that covers this identity directly, the phenomenon reminds me structurally of Midy's Theorem, where a repeating decimal of even period leads to a sum of its halves being composed of 9s. Here, the square of a number with repeated digits contains its own parts in such a way that the sum of a suitable split of the result gives back the original number, almost like a "reverse-Midy" identity — applied to integers instead of periodic decimals. Your example with 22222² is striking, and the logic in your attachment outlines a family of such patterns. It feels like it deserves more exploration — maybe a generalized formulation could be stated in terms of decimal symmetry and modular behavior.

Best,

Charles Kusniec

ckus...@gmail.com

https://orcid.org/0009-0006-1146-7847

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[Ed Pegg]

The cube root of 702331940521262213443072  (no 8's) is   
88888888   
Also, 70233194+05212622+13443072 = 88888888   

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[Daniel Mondot]

The family where you get back the original number matches: A145875

Interestingly, it was just edited 3 days ago by Neil, adding new terms from Gupta.

Daniel.

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[Sam Khan]

My son had the idea of subtraction too. So 11^2 = 121. 12-1=11.

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[Robert Munafo]

Numbers like 22222 are called Kaprekar numbers. See www.mrob.com/pub/math/numbers-7.html#la45 and  www.mrob.com/pub/seq/kaprekar.html for lots of info and links. Note that 45 works for squares, cubes and 4th powers which is why I call it "the quintessential Kaprekar number".

[Daniel Mondot]

Interesting tidbit about A145875 (the sequence where, after squaring n, and adding the 2 half of n^2, we get n back):

If we divide each term by the largest unit repdigit (1, 11, 111, etc...) i.e. we reduce it to 1 digit, we end-up with a repeating sequence of 15 terms: 1,9,5,9,9,7,9,2,9,9,4,9,8,9,9.

Daniel.

On Fri, Mar 28, 2025 at 11:00 AM Robert Munafo <mro...@gmail.com> wrote:

Numbers like 22222 are called Kaprekar numbers. See www.mrob.com/pub/math/numbers-7.html#la45 and  www.mrob.com/pub/seq/kaprekar.html for lots of info and links. Note that 45 works for squares, cubes and 4th powers which is why I call it "the quintessential Kaprekar number".

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