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[ July 14, 2002 ]
A very unique palindrome 549945

549945 is the only fixed-point palindrome
for the Kaprekar process for base-10 numbers
containing fewer than 16 digits.

A neat palindromic fact spotted by Terry Trotter
when he visited Walter Schneider's webpage at
MATHEWS : Kaprekar Process
( now alas a broken link! )

This Kaprekar process is illustrated as follows :

549945 995544445599 549945

[ July 19, 2002 ]
No person is better placed to explain the topic
than Terry Trotter (email) himself.

  A Palindrome à la Kaprekar  

Here is an interesting palindrome: 995544445599.
To see why, we must first introduce a popular idea that has
fascinated mathematicians and school students alike for sometime.
It's called Kaprekar's Ordered Subtraction Operation (OSO).
The OSO may be described thus:

Start with a four-digit number whose digits are not all equal,
arrange the digits in descending and ascending order, subtract
and repeat the process. Then the process terminates on the
number 6174 after seven or fewer steps.
The amazing thing here is that 6174 always results, no matter
what the initial 4-digit number might be (excepting those consisting
of only one digit, like 3333, of course).

As one does the OSO process with numbers of more or fewer digits,
different outcomes can result. With 3-digit numbers, the constant
final outcome is always 495. I call that a terminator. With 2-digit
numbers, the following five-term cycle is the only outcome
09 81 63 27 45 09

But things really become interesting when using numbers of 5 or
more digits
. Here cycles and terminators can occur for a given case.
(See Walter Schneider's website for all the outcomes
up to 15-digit numbers in base 10.
)

A quick examination of that table reveals the fact that there
is only one palindrome. It occurs for 6-digit numbers. 549945 is one
of two terminators possible, accompanied by one cycle of seven terms.

By now it should be obvious how the 12-digit palindrome above
was created. It is a number with the interesting property
that can be stated as:

A palindrome with an even number of digits such that if it is
separated into 'halves', the positive difference of the two parts
is itself a palindrome.

995544445599 995544445599 549945.

Now, I can only but wonder if there are other palindromes
that share this special property.

ps1. regarding your doubt about fixed points...
I just call such things terminators or self-producers.
6174 is often called Kaprekar's Constant.

ps2. For every Kaprekar Palindrome, as we might call
those numbers with the split-half property, there is always
a companion. Just reverse the 'halves', like so:
445599995544
trivial, perhaps, but necessary for completeness.




A000135 Prime Curios! Prime Puzzle
Wikipedia 135 Le nombre 135














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