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I'd like to hear insights and theory of the mirror numbers and their possible significance in mathematics and geometry. With mirror numbers I mean these four examples:

432 -> 234 123 -> 321 153 -> 351 987 -> 789 

Sum of 432 & 234 is 666 and sum of 153 & 351 is 504, which are famous numbers from historical perspective, namely from Pythagoras, Plato, Archimedes and Revelation of John.

Supplementing questions arose on a chat with Dan:

1) How to determine if a number x can be represented as n + rev(n)?

2) How to determine possible n & rev(n) for number x?

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    Seems to be an exact match, @Robert Thanks for reference and heads up for 5 years old issue.2017-04-28

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This is not terrifically profound, but if $y$ is the mirror of $x$ in base $b$, then $x \equiv y \mod b-1$, while $x \equiv \sigma y \mod b+1$ where $\sigma = 1$ if $x$ and $y$ have an odd number of base-$b$ digits and $-1$ if they have an even number of base-$b$ digits.

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    let us [continue this discussion in ch$a$t](htt$p$://chat.stackexchange.com/rooms/6362/discussion-between-dan-brumleve-and-phpgae)2012-11-07
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Such pairs of numbers have no intrinsic mathematical significance (although related numbers are studied; see palindromic numbers) due to the arbitrary choice of base $10$. I don't know of any geometric connection but I would expect it to be very subtle, not as direct as what you are suggesting. The fact that there is both a 2-3-4 triangle and a 6-6-6 triangle is a coincidence emboldened by the choice of a relatively large base.

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    Base 2 would be a better place to start with more fundamental meaning. The smaller the base the better the theorem. The historical connections are interesting but consider the context: 60 and 360 are used as bases because of their smoothness, not because of any digital consummations.2012-11-07
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Subtract the small mirror from the large mirror & divide by 9. It's fun and very symbolic. Mirrors without and within.

For example:

2559591 - 1959552 = 600039

600039 / 9 = 66671

8883 - 3888 = 4995

4995 / 9 = 555

5832 - 2385 = 3447

3447 / 9 = 383

25367473251-15237476352 = 10129996899

10129996899/9 = 1125555211

Try things like year of birth & another year:

5891 - 1985 = 3906

3906 / 9 = 434

7102 - 2017 = 5085

5085 / 9 = 565

565 + 434 = 999

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    There is a rule of nine. Transpose digits of the number in any base. Substract and result should be divisible by base - 1. Above mirror feature is a special case of such much more general feature.2017-08-07