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Questions tagged [palindrome]

Palindromic number or numeral palindrome is a number that remains the same when its digits are reversed. Questions involving palindromes, or mathematics related to them such as the Lychrel process or Scheherazade numbers.

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Probability of picking $n$ a palindrome

I am struggling to see how to solve this question. My confusion stems from the Less probable, for n plain drones the equation I derived was $9\times10^{\frac{n+1}{2}-1}$ this is under the assumption ...
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Circular Happy Palindromic Primes

$(1)$ A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a ...
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Is there a palindrome prime $p>3$ that is also palindrome in base $\ 5\ $?

A prime $\ p\ $ is called palindrome, if the digits in reverse order give the same prime. For bases $\ b=2,3,4\ $ , there are large examples of palindrome primes that are also palindrome in base $b$ , ...
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Can the product of $n$ positive integers, where $n \gt 5$ in A.P. be a palindrome?

Reading the question can the product of four positive integers in A.P. be a square?, also made me question whether the product of $n$ positive integers, where $n \gt 5$ in arithmetic progression be a ...
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Palindromes in three consecutive bases - How do I effectively extend the proof to more lengths of digit cases?

Summary I do not know how to solve the problem of finding all numbers palindromic in three consecutive number bases generally, I've since split it into countably many subproblems, each asking to ...
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nth term of the series containing one 9 then two 9 and so on [closed]

nth term of the series 9, 99, 999, 9999. Is there any formula for it? 9 99 999 9999 . . . .
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palindrome number

How to find the $n$-th term of the series. 1 11 101 1001 10001 100001 1000001
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Potential error in the paper: “Every positive integer is the sum of three palindromes”

Maybe some of you know of the nifty trick that every number can be split into three palindromic numbers that add up to said number. As described in the article right here, this works so far so good, ...
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Are all Fibonacci words uniquely represented as concatenation of two palindromes?

Suppose Fibonacci word sequence is a word sequence defined by the following relations: $$\phi_1 = «0»$$ $$\phi_2 = «01»$$ $$\forall n > 2 \text{ } \phi_n = \phi_{n - 1}\phi_{n - 2}$$ Let’s prove, ...
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Palindrome vs Level of Palindrome

The palindrome, example: $131$, $82728$, $55655$. But from the palindrome maker algorithm say: If $17$ isn't palindrome you must additive by reverse of them $33$ is say $P(1)$ palindrome $38$ is ...
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Even number not the sum of two base $2$ palindromes

What's the least positive even number not a sum of two base $2$ palindromes? I've checked and it must be over $100$ since all up to $100$ are such sums. [Or, which to me seems unlikely, are all even ...
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Find the number of words recursively such that there is no palindromic suffix

Given a set of distinct characters $\{a_1, a_2, \cdots , a_S\}$ and a number $N$, find the number of words of length $N$ that can be formed using these letters (repetition allowed) such that there is ...
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Proof verification of the language of all palindromes as being context-free

Consider that the language L of all palindromes over $\Sigma = \{0,1\}^*$ is not context-free. The following is my attempt at a proof by contradiction. I am new to proof writing and I am wondering ...
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Is the palindrome-prime factor of $p-1$ always larger than that of $p+1$?

Suppose, $p\ge 7$ is a palindrome-prime , the largest prime factor of $p-1$ is a palindrome-prime and the largest prime factor of $p+1$ is also a palindrome-prime. Must the prime factor of $p-1$ ...
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Octal palindromes with even number digits are all composite numbers? [closed]

I want to know whether octal palindromes with even number digits (11 or 1221, but not 121) are all composite numbers, and a general proof if so or a counterexample if not.
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$p$ and $6p+1$ both palindrome - primes. Is $(131/787)$ the only example?

$131$ is a palindrome prime as well as $787$ , moreover $6\cdot 131+1=787$. Are there further examples for a palindrome-prime $p$, such that $6p+1$ is a palindrome-prime as well ? It is clear that ...
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When are $a+b$ and $ab$ palindromic for integers $a,b$?

This question came up when I was discussing Ex 1.23 of An Introduction to Analytic Number Theory (Apostol) with a user. For positive integers $a,b$, when are the values of $a+b$ and $ab$ ...
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How many palindromes are there in the range $0000$ to $9999$?

Apologies if this is a basic question, but my math is weak and this is something I've been wondering lately. This viral Facebook post repeats a long-standing myth that states: If a thief forces ...
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How to make a palindromic table where each row and column is a palindrome?

I have a set of digits. For example, 3 1s, 2 2s, 10 3s, ... With this available digits how do I create a palindromic table of fixed (nxm) matrix where each row and column is palindromic? For a ...
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How many palindromes of length 5 can be formed that contain 7 or 8?

I'm trying to figure out how many palindromes of length 5 can be formed that contain 7 or 8. My reasoning is as follows: The pool to choose from is $\{0,1,2,3,4,5,6,7,8,9\}$ and there are three ...
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Patterns in solutions to simultaneous palindromes in two number bases

Edit: Decided to add visual previews of patterns instead of extracting all the polynomials. Perhaps you have seen something that behaves similarly somewhere else? For a given $b,d\in\mathbb N$, how ...
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Palindrome multiple of prime

Let $p$ be a prime number less than $2000$. Prove that there exists a multiple of $p$ that is a palindrome and has at most $450$ digits. We can assume that $p>9$ (in the case $p\leq 9$, $p$ is a ...
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How many $4$-digit palindromes are divisible by $3$?

How many $4$-digit palindromes are divisible by $3$? I'm trying to figure this one out. I know that if a number is divisible by $3$, then the sum of its digits is divisible by $3$. All I have done is ...
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Find all the prospective numbers with maximum $N$-digits [closed]

Find all numbers with maximum $N$-digits (could be less than $N$-digits), when added to its reverse form (ex: reverse form of $123$ is $321$, of $452$ is $254$), would become a palindrome number (...
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The quotient of two palindromes

Besides the multiples of 10, are there infinitely many positive integers which are not the quotient of two palindromes? Is 12 such an exception?
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Substitution in Palindromic Polynomial

Given a palindromic polynomial $p(x)$ of degree $2n$, where the coefficients are $a_{i}=a_{2n-i}$. It is known that there exists a polynomial $q$ with $$ x^n q(x + 1/x) = p(x) $$ I'm looking for an ...
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Why is it so hard to find numbers with some palindromic properties?

Lets start by defining palindromes and then looking at some problems; We can define the set $P_d$ as set of all palindromes with $d$ digits. If $p\in P_d$, then $p=(x,y)$, where $x$ is the ...
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Infinitely many simultaneous palindromes in “exponentially close” number bases?

I've observed regular patterns that suggest that there are infinitely many numbers which are simultaneously palindromic in $n$ number bases of form: $$ b_1,b_2,b_3,\dots,b_n={x^{y_1},x^{y_2},...
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Most palindromic prime numbers

A prime number is the most palindromic prime if it has more nontrivial palindromic representations across number bases than all primes below it. Nontrivial representation means we are not ...
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Predict gaps between 9 digit 2-palindromes?

$p^2$ $\text{(or 2-palindrome) definition: }$ I call a palindrome $n$-palindromic or $p^n$ if it is palindromic in $n$ consecutive number bases. For example, $10$ is $p^2$ since it is palindromic in ...
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Unexpected pattern in consecutive 7 digit “double” palindromes?

Why does this branching structure appear in consecutive 7 digit "double" palindromes? Dots at the end of the rows indicate that I excluded following values for that row (because of the character ...
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Find palindromes in two consecutive number bases?

Can we generate all numbers which are palindromic in two consecutive number bases $(b, b+1)$ and have $(2d+1, d\in\mathbb N)$ digits when written in their palindromic bases? (I strongly ...
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Represent all palindromic numbers across number bases based on digit count

I've been looking at plots/graphs of palindromic numbers across number bases, and noticed that we should be able to represent all $d$ digit palindromes across number bases $y$, with some expression $...
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Palindromes in multiple bases

I noticed when listing out palindromes in bases $2$ and $3$ that they seem not to share any palindromes (other than trivial single-digit palindromes). However, when I tried to prove this, I couldn't ...
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Sum of the first $n$ palindromes

We put together a problem to be solved programmatically, and we know at lower numbers there is a solution to this problem. How would we go about proving whether the below problem has an answer, as our ...
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How to calculate the palindrome of a given amount of characters?

A palindrome is a set of characters which equal the same thing forwards and backwards, for example; abccba For a set of a given amount of characters (we can use 9 as an example) how would you ...
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Write the even perfect number $8128$ as sum of three palindromes

I would like to know, well from your calculations with a computer and your explanation of your computational method, or well with a theoretical argument how to show that an even perfect number, for ...
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Recursively palindromic primes. Are they rare birds?

A prime number is a number larger than 1 which only positive divisors are itself and 1. Examples: 3,5,11. A number is palindromic in a base $b$ if when written with digits in that basis $d_1d_2\...
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239 views

Square of palindromic number

The other day I thought of this question: Is it possible to find a set of palindromic numbers such that the square of them is itself palindromic? In other words: $A = \{a : a^{2}=b$ where a,b are ...
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Sequences of numbers palindromic in 2 consecutive number bases?

Lets look at sequences of numbers that are a palindromic number in two consecutive number bases $b$ and $b+1$, where $b\ge2$ of course. (And also ignoring trivial one digit palindromes.) I would ...
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334 views

“Reverse base” Riemann Zeta Function?

Riemann Zeta Function: $$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\frac{1}{5^{s}}+\frac{1}{6^{s}}+\frac{1}{7^{s}}+\frac{1}{8^{s}}+\frac{1}{9^{s}}+\frac{1}{10^{s}}+\dots$$ "...
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How do you proof that the simply periodic continuous fraction is palindromic for the square root of positive primes?

I have formulated this question based on the initial curiosity and further investigation of the topic posted here: Identity and possible generalization of the reflective periodic continued fractions ...
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Palindromic Numbers - Pattern “inside” Prime Numbers?

EDIT: rewritten and reduced entire post to present things more clearly. I'm asking how to calculate the next element(s) in the sequence, located at the end of this post. Introduction - ...
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Expanding Palindromes

I couldn't find anything about this on the internet and was wondering if there was any information or work done on an idea regarding expanding palindromes? I'm defining expanding palindromes as A ...
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1answer
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Why are multiples of 11 palindromes?

One thing I noticed is that for any integer $-10<a<10$, $11a$ is always a palindrome. I'm assuming this is because 11 is the first row of Pascal's triangle. For that same reason, for any ...
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How to get palindrome of a number with specific base [closed]

I have problem while reversing and checking palindrome for a number of specific base (other than 10). For example:- let's take 87. The Palindrome number is found as follows: ...
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How many 8-digit palindromes are prime? [closed]

An integer is said to be palindrome if it reads the same forward or backward. For example, the integer 14541 is a 5 digit palindrome and 12345 is not a palindrome. How many 8-digit palindromes are ...
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Find the smaller palindrome that is divisible by $3,5,11$

It seems to me that there are only two ways to solve the problem: First-Use divisibility rules to select last digits, and work my way from there Second-Write out all multiples of 165 since 3, 5, and ...
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Can a number be palindrome in 4 consecutive number bases?

$4$ consecutive bases? Are there numbers that are a palindrome in $4$ consecutive number bases? Note that I'm not counting one digit palindromes, since one digit numbers $x$ are trivially ...
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Which number base contains the most Palindromic Numbers?

Plotting Palindromic Numbers I made a script that checks numbers through number bases and plots a black pixel if the number is a palindrome in the corresponding base. If we check the first $256$ ...