Talk:Palindromic number

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What about 196?

I assume you're talking about the status of 196 as the smallest (suspected) Lychrel number? It's discussed in that article, although I've added a link to it here in the "see also" section. Chuck 15:33, 31 October 2005 (UTC)

[edit] Infinitely many

I know it's obvious, but I think it should be stated that in any given base there are infinitely many palindromic numbers. So I've added that in the top section. I don't really have any feelings about where in the article it should go, though, so move it elsewhere in the article if you want. Hammerite 01:05, 2 March 2006 (UTC)

[edit] Percentage

What is the percentage of all numbers that are palindromic?

0%--as the number of digits grows larger, the fraction of those numbers which are palindromic grows smaller, approaching 0% as the number of digits goes to infinity. For example:
Number of digits Range of numbers Total numbers in range Palindromic numbers in range Fraction of palindromic numbers in range Cumulative range Cumulative total numbers Cumulative palindromic numbers Cumulative fraction
1 0-9 10 10 1 0-9 10 10 1
2 10-99 90 9 0.1 0-99 100 19 0.19
3 100-999 900 90 0.1 0-999 1000 109 0.109
4 1000-9999 9000 90 0.01 0-9999 10000 199 0.0199
5 10000-99999 90000 900 0.01 0-99999 100000 1099 0.01099
6 100000-999999 900000 900 0.001 0-999999 1000000 1999 0.001999
To be more general, the number of palindromic numbers in base b with n digits is:
\begin{cases} b, & n=1 \\ (b-1)b^{\left \lfloor \frac{n-1}{2} \right \rfloor}, & n>1 \end{cases}
where \left \lfloor x \right \rfloor is the floor function (the floor function in that formula is the exponent of b, even though that's not too clear the way it's represented here).
And the fraction of numbers in base b with n digits which are palindromic is:
\frac{1}{b^{\left \lfloor \frac{n-1}{2} \right \rfloor}}
The fraction of numbers in base b with n or fewer digits which are palindromic is:
\frac{b^{\left \lfloor \frac{n+1}{2} \right \rfloor} + b^{\left \lfloor \frac{n}{2} \right \rfloor} -1}{b^n}
which approaches zero as n goes to infinity.
This is all original research, of course, so it can't go in the article itself unless an outside source can be found for it. Chuck 15:31, 8 March 2006 (UTC)

[edit] phrasing

I don't understand what this means: "Buckminster Fuller numbers as Scheherazade numbers in his book Synergetics, because Scheherazade was the name of the story-telling wife in It is fairly straightforward to appreciate that in any base there are infinitely many palindromic numbers, since in any base the infinite sequence of numbers written (in that base) as 101, 1001, 10001, etc. (in which the nth number is a 1, followed by n zeroes, followed by a 1) consists of palindromic numbers only."

Should it be a bulleted statement? Is it actually two facts? If it is two facts, should the second fact be bulleted as well? I would propose:


  • Buckminster Fuller numbers, such as Scheherazade numbers, from his book Synergetics

It is fairly straightforward to appreciate that in any base there are infinitely many palindromic numbers, since in any base the infinite sequence of numbers written (in that base) as 101, 1001, 10001, etc. (in which the nth number is a 1, followed by n zeroes, followed by a 1) consists of palindromic numbers only.


--Amanaplanacanalpanama 00:12, 28 August 2006 (UTC)

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