Talk:Palindromic number

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)

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)

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)

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)

Convertion

I added a method of converting non-palindromic numbers to palindromic numbers. Hopefully the informtion is useful, but if it should be deleted, then I'm sorry...Littleghostboo[ talk ] (Win an argument and leave your mark in history.) 09:08, 29 April 2007 (UTC)

Is convertion a math word? I didn't find it on dictionary.com or in wikipedia. Should this heading be conversion instead?

Section Deletions

I've removed the two miniscule sections on other bases and other languages. One is clearly superfluous, while the other is fixed on a subject that may belong here but needs expanding if it does. I will also be momentarily adding some to the subject of other bases. It's my own work, in OEIS, that I'll be referencing, so I'm not sure if I'm following proper procedure to include it.Julzes (talk) 22:24, 9 March 2011 (UTC)

My own work available in OEIS

Rather than include it myself, I wish to simply point to my own sequences in the OEIS in the event somebody else may feel discussion of them--multi-base palindromes--a worthy addition. I only have 60-some approved sequences, so they may be found rather easily with a search of 'author:Merickel'. Discussion of other material that can be found at worldofnumbers.com (I am not affiliated) might also be considered.Julzes (talk) 02:11, 1 December 2011 (UTC)

Scheherazade numbers

What is Buckminster Fuller's definition of Scheherazade numbers? It does not seem to be a straigtforward synonym of palindromic numbers. I cannot see a definition in his book from the little I can see in gbooks snippets, but the examples given lead me to believe he means a number that is palindomic by digit groups and contains all the prime factors up to some prime p and no prime factors above p

1×2×3×5×7×11×13 = 30-0-30

1×2×3×5×7×11×13×17 = 510-510

(1×2×3×5×7×11×13)^2×17 = 153-306-153

(1×2×3×5×7×11×13)^2×17×3 = 459-918-459

There is also what he calls a seven-illion Scheherazade number, and a fourteen-illion Scheherazade number, which uses the first 15 primes, but I can't see what the actual number is. Ironically, if that is the definition, 1001 does not fit it! Perhaps the definition should be modified to the prime factors have to be contiguous (but can start anywhere). SpinningSpark 18:55, 23 November 2015 (UTC)

Palindrome Days

Mention Palindrome days, like 20200202 (2020-0202). Jidanni (talk) 23:43, 22 January 2020 (UTC)

Different types of palindromes

There are different types of palindrome numbers. Should there be a new section on the page for the same or the a new page whole together, Types of Palindrome numbers. I came across 4-5 types related to palindromes that are not mentioned here on this page.Adamsamuelwilson (talk) 18:28, 28 April 2021 (UTC)

Binary palindromes

@David Eppstein: Why is (sequence A057148 in the OEIS) an off-topic sequence?. It's titled as "Palindromes only using 0 and 1 (i.e., base-2 palindromes)" which seems to me to be precisely the topic it was attached to. SpinningSpark 23:15, 5 November 2021 (UTC)

Because it's the sequence of numbers whose decimal representations have only zeros and ones, in a sentence about numbers whose binary representations are palindromic. It's relevant for the general topic of palindromic numbers but not for binary palindromes. —David Eppstein (talk) 23:17, 5 November 2021 (UTC)

So why does the title say base-2 palindromes if this is a decimal sequence? It's the same digits either way in any case. SpinningSpark 08:53, 6 November 2021 (UTC)

OEIS:A057148 has the comment "For each term having fewer than 10 digits, the square will also be a palindrome". This applies to decimal but not binary. For example, 101 in binary is 5. 5² = 25 is 11001 in binary. In decimal, 101² = 10201. PrimeHunter (talk) 00:22, 7 November 2021 (UTC)

Yes, I had spotted that myself, but does a comment override the definition in the title? SpinningSpark 08:20, 7 November 2021 (UTC)

OEIS generally uses base 10. "Palindromes only using 0 and 1 (i.e., base-2 palindromes)." would be an odd title if "base-2 palindromes" was the definition and not just a parenthetical remark. I think it's about an alternative way to read the numbers, not the actual definition. OEIS:A118594 "Palindromes in base 3 (written in base 3)" is how to write a definition properly if it doesn't use the default base 10. PrimeHunter (talk) 20:08, 7 November 2021 (UTC)

"Palindromes in base 2" would be exactly the same sequence, which explains why that title does not exist. SpinningSpark 14:28, 3 December 2021 (UTC)

Features of palindromic numbers

What are the features of palindromic number?? 2402:3A80:1BA4:4654:578:5634:1232:5476 (talk) 09:31, 8 December 2021 (UTC)

Palindromic numbers with an even number of digits

Unless I'm missing something, there's no mention of the fact that any palindromic number with an even number of digits is divisible by 11. Kostaki mou (talk) 18:12, 7 January 2022 (UTC)

I will add that to the page. Adamsamuelwilson (talk) 11:05, 11 July 2023 (UTC)

It's mentioned in the lead of Palindromic prime. We can also mention it here but I moved it out of the lead and reformulated it.[1] More generally, a base b palindrome with an even number of digits is divisible by b + 1. If n is odd then bⁿ + 1 is divisible by b + 1. It follows from that. PrimeHunter (talk) 11:43, 11 July 2023 (UTC)

Unique property of 12 and 13?

The square of 12 when its order is reversed and the positive square root taken (which in the case of 12 is 21), it seen that it is the reverse of the original number. This property also applies to 13. What is the name for this curious feature and are there any other numbers that have it? Macrocompassion (talk) 08:21, 4 October 2023 (UTC)