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For introduction, Ethiointegers are integers which get reversed when multiplied by another number.

For instance,

2178 * 4 = 8712

1089 * 9 = 9801

I couldn't find such numbers, even by another name anywhere else except in one journal which I think is not electronically accessible.

My question: Has numbers like this been treated anywhere? If so, what interesting properties do they have?

[E.g. One property I consider interesting: sumofdivisors(1089) = 1729 = 1728 + 1 (Hardy's taxi number 12^3 + 1^3, more about this sum ), also 1728 is a rearrangement of 2178!]

[Added later] Eulertotient(2178) = Eulertotient(1089)

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There were similar in spirit questions, for example, mathoverflow.net/questions/23512. Why do you think this property is worth having a name?! –  Wadim Zudilin Jun 12 '10 at 10:47
The question has attracted one good answer which helps to explain why this is not of interest to research mathematicians (unless they also happen to be avid puzzle fans). Hence I am voting to close. –  Harald Hanche-Olsen Jun 12 '10 at 11:46
I don't believe that this question should be closed. Hardy's comment should be taken about as seriously as his comment that number theory has no practical applications, i.e., not seriously at all. Problems about representations of integers in specific bases have traditionally been unfashionable because any questions about them tend to be either trivial or intractable, but there's no a priori reason this has to be the case. It's quite possible that something mathematically interesting could emerge. At any rate, it's entirely reasonable to ask if anything is known about such numbers. –  Timothy Chow Jun 13 '10 at 1:29
If you look at the OEIS list of sequences that satisfy this property, they look very boring in the following sense: they're all of the form XXXXX...X, periodic, with X either 109999...9989 or 21999..99978 (with zero 9's allowed in the middle of X). Is it true that any such number is of this form? If so then they're officially boring :-) On the other hand if one is known that isn't of this form then maybe they just got interesting :-) There's a concrete question that may well be easily resolved and would give some indication of whether there is any maths in these numbers. –  Kevin Buzzard Jun 13 '10 at 7:20
I feel that MO becomes a numerology site: there are several questions which try to mystify certain curious (but not more!) observations about numbers. Plenty of instances at the OEIS are far-fetched, and for each such instance one can ask a question... I would be happy to see motivation. –  Wadim Zudilin Jun 13 '10 at 8:57
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1 Answer

up vote 12 down vote accepted

These unnamed numbers were famous enough to make it to A Mathematician's Apology. After mentioning what you wrote above Hardy writes:

These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals much to a mathematician. The proofs are neither difficult nor interesting—merely a little tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and the proofs, which are not capable of any significant generalization.

These numbers are listed in OEIS, and there it is mentioned that

Theorem (David W. Wilson): If reverse(n) = k*n in base 10, then k = 1, 4 or 9.

Off the top of my head I can make the conjecture that none of these numbers contain the digit 3, for example, but I don't know of any interesting mathematical ideas that could be involved in the proof (this doesn't mean anything though, and is of course very subjective).

Edit: To answer your question about places where these numbers have been considered, I will mention this article by Lara Pudwell. The author's views are opposite to Hardy's, and she describes a few related problems and generalizations.

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And after all that they remain unnamed?! +1 –  Wadim Zudilin Jun 12 '10 at 11:43
There are numerous articles on these, but they are always introduced as "a curiosity", "a previously made observation" or "an interesting phenomenon" and then refer to authors who previously wrote about them. –  Gjergji Zaimi Jun 12 '10 at 11:52
OK, maybe this should get closed as unsuitable to the purpose of this forum, but I can't resist mentioning an example that one sometimes comes across, just because it's maybe gaudier than the ones mentioned: $$ 123456789 \times 8 = 987654312. $$ –  Michael Hardy Jun 12 '10 at 19:24
This is a near miss and I knew this and others alike in primary school. –  Unknown Jun 12 '10 at 19:29
Wow, I am excited to see the paper. Many thanks. I now understand how much collaboration works to one's advantage –  Unknown Apr 6 '11 at 17:26
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