# 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.

52 questions
1answer
34 views

### 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 ...
0answers
76 views

### 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 ...
1answer
148 views

### 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 $b$ , ...
0answers
46 views

### 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 ...
0answers
36 views

### 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 ...
1answer
34 views

### 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 . . . .
1answer
37 views

### palindrome number

How to find the $n$-th term of the series. 1 11 101 1001 10001 100001 1000001
1answer
91 views

### 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, ...
1answer
83 views

### 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, ...
1answer
31 views

### 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 ...
1answer
36 views

### 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 ...
1answer
43 views

### 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 ...
1answer
77 views

### 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 ...
1answer
85 views

### 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 $p-1$ ...
1answer
35 views

### 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.
1answer
176 views

### $p$ 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 ...
1answer
173 views

### 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 $ab$ ...
1answer
97 views

### 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 ...
1answer
245 views

### 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 ...
2answers
57 views

### 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 ...
0answers
163 views

### 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 ...
1answer
114 views

### 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 ...
2answers
2k views

### 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 ...
0answers
81 views

### 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 (...
0answers
167 views

### 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?
2answers
76 views

### 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 ...
0answers
61 views

### 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 ...
1answer
48 views

1answer
205 views

### 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 ...
1answer
1k views

### 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 - ...
1answer
88 views

### 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 ...
1answer
795 views

### 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 ...
1answer
94 views

### 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: ...
2answers
876 views

### 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 ...
1answer
93 views

### 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 ...
1answer
943 views

### 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 ...
1answer
352 views

### 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 $256$ ...