Questions tagged [palindrome]

A palindrome is a number or any other sequence of characters which remains the same when it is reversed (read backwards). Questions involving palindromes, or mathematics related to them such as the Lychrel process or Scheherazade numbers.

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Can finite sequence of $\pi$ starting from first digit of $\pi$ $3$ be palindromic?

To clarify my question consider the following transcendental number: $5.123215314159...=5.123215+\pi\cdot10^{-7}$ note that :finite sequence starting from first digit $5$ that is palindromic is $...
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The palindrome counting function

Let $b \geq 2$ be a positive integer and consider the function $f_b : \mathbb{N}^+ \to \mathbb{N}^+$ given by $$f_b (n) = |\{ k \in \mathbb{N}^+ : k \leq n \mbox{ and } k \mbox{ is palindromic in base ...
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Counting problem about Palindromes

Consider the set of four digit sequences $d_1d_2d_3d_4$, where $d_i\in\{0,1,\ldots,9\}$. (a) What is the number of all four digit sequences, which contain no palindromic subsequence. For example, the ...
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If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?

Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers or positive integers. Recall that a Descartes number is an odd ...
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Understanding a statement about palindromic numbers

I have the following statement: "Palindromes starting with $n$ such that the sum of the digits of the product of the factorial of $n$ and reverse of $n$ is equal to the center digit." And I ...
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Can we find a palindrome for more than $5$ factors?

For positive integers $a,b$ and $k$ , define $$p(a,b,k):=\prod_{j=1}^k (a(j-1)+b)$$ That is the product of the first $k$ numbers in the arithmetic progression $an+b$ starting with $b$ Can $p(a,b,k)$ ...
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Cycles in modified reverse-and-add process

Consider repeatedly adding a reverse (of digits) of a number to itself, until a palindrome is reached. Numbers that never reach a palindrome (diverging to infinity), are called Lychrel numbers. Let's ...
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Conjecture related to Lychrel numbers

My conjecture is as follows: Take any number. It will become a palindrome eventually through the same reversal process used for Lychrel numbers except if the term (first term is excluded) starts with ...
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Turning an informal proof about palindromes into a formal proof

No. of dots: $k$ No. of slots: $n$ If the dots are placed in every combination within the slots, how many palindromes will there be? The dots cannot be superimposed on each other, which means $k \lt n$...
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Number written forward plus backward =100?

Is there a Percent where when written forward plus backward equal 100? Example: XY + YX = 100 (where those variables aren't multiplied together but stand in for ...
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Can Peano arithmetic define the set of base 10 palindromic numbers?

A base 10 palindrome is a number that when written in base 10 and the digits are reversed, gives the same number. I am wondering, can Peano arithmetic define the set of base 10 palindromes? I don't ...
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Is $404$ a palindrome in base negative $31$?

I was browsing on a website and I accidentally clicked on a link (Here is the link, but it may not show the same thing). The following was written there: $404$ is also a palindrome in base negative $...
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A conjecture about binary palindromes and arithmetic derivatives

Corrected question. From the sequence of binary palindromes A006995 (eg. 1001001001001) the sequence of possible gaps between consecutive palindromes contain the elements: ...
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Find all four-digit numbers $ \overline {abcd}$ where $a + b = c + d$ which are the sum of two palindromic numbers.

We say that a four-digit number $ \overline {abcd}$ is balanced if $a + b = c + d$. Find all the balanced four-digit numbers that can be expressed as the sum of two palindromic numbers. I proved that ...
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Project Euler exercise 4: largest palindrome made from the product of two 3-digit numbers

Consider the 4th problem of the Project Euler: A palindromic number reads the same both ways. The largest palindrome made from the product of two $2$-digit numbers is $9009$ = $91 \times 99$. Find ...
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Find the n-th base 2 palindrome

So I want to compute the n-th base 2 palindrome, where n is a number less than or equal to 5000, and we can pick it. So we know that in base 2 only odd numbers can be palindromes. Okay so that saves ...
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7 digit palindrome problem [closed]

This is my problem: From {1,2,3, ..., 9} How many palindromes of length 7 are there, where each digit can appear at most twice
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Number of configurations of dots and loops without mirrors

The equation for number of ways to put $b$ loops around $b$ of $a$ dots is $$a \choose b$$ How can one modify this so that all palindromes/mirrors are excluded? This is a continuation of this ...
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How many 5-digit palindromes are divisible by 9?

What I already have, Palindrome in form XYZYX, where X can’t be 0. Divisibility rule of 9: sum of digits is divisible by 9. So, we have 2(X+Y)+Z = 9M. The first part is divisible by 9 if and only if ...
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Prove that if $z^n$ is a palindrome for some $n>0$, then $z$ is also a palindrome for any alphabet E.

Prove that if $z^n$ is a palindrome for some $n>0$, then $z$ is also a palindrome for any alphabet E. Here's a proof of the statement above: Let $w = x^n$. If $n = 1$, then the result is trivial. ...
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How to count number string that read the same in forward and backward

Suppose there is a number string that have 10 digit and I want find how many combination that read the same in forward and backward I have tried 10x9x8x7x6 but it that the case for distinct case like ...
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What is the proof behind this pattern?

Let me get to the point. We see that 121 is a palindrome, also 12321 is a palindrome. If we were to add up the digits in each palindrome, for example, 121 we get 4. If we square root four, the root is ...
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Probability of a palindrome

$5$ $A$'s and $6$ $B$'s are arranged in a row. Find the probability that a choosen arrangement is palindrome. I tried out all the ways a palindrome could be formed. This was favourable outcomes. All ...
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Palindromic divisibility and primes

Inspired by this question: Define odd-palindroming $X$ in base $b$, or $OP(X,b)$: Take an integer $X$ and write it in base $b$. Then, reverse its digits and concatenate the reversed digits to the end ...
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How to find the count of d digit numbers such that N is divisible by Rev(N)?

How to find the count of d digit numbers such that N is divisible by Rev(N)? For example if d = 4, then answer = 2 Since There are only 2 such numbers 8712 and 9801 8712 % 2178 = 0 9801 % 1089 = 0
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How to represent N as minimum number of palindrome numbers?

How to represent N as minimum number of palindrome numbers? for eg 127 = 121+6 for eg 200 = 191 + 9
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Observation and conjecture on Lychrel numbers

A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number that remains the same when its digits are reversed. Consider a number $n>0$ in base $b\geq2$, where it ...
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What is the probability of the following license plate containing palindromes?

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the ...
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Why does every 3-string, composed of two letters have exactly 3 palindromic substrings?

Why does every 3-string, composed of two letters have exactly 3 palindromic substrings? for example: aab has a, b and aa aba has a,b and aba I have been doing some practice questions for an upcoming ...
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Proof about the recursive definition of a palindrome

This problem is taken from "Mathematics for Computer Science" (Lehman, Leighton, Meyers, 2018). 1. Definitions First, some relevant definitions. 1.1. String The set $A^*$ of strings over alphabet $...
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Count of palindrome numbers in the range [a,b]

Can we find the count of palindrome numbers in the range [a,b] by directly using a formula. for eg in range [8,12] there are 3 palindrome numbers (8, 9, 11)
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How can I prove that all strings in {0, 1} * that contain any palindrome of length at least 6 as a substring is regular?

Because palindrome is irregular, I do not know how to prove above, even if I know for example, {0^n1^n|n>0} is irregular but it is a subset of 0*1*, which is regular. I don't think I can use, for ...
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Palindromic Number Problem

Suppose $a$ be a 28-digit palindromic number. Given that $a$ is a multiple of $13$ and all the digits except the 13th, 14th, 15th, and 16th are $1$. Let $A, B, C, D$ be the 13th, 14th, 15th, and 16th ...
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How can I show that even digit palindromic numbers are divisible by $11$?

I do not really have much background knowledge in number theory, so maybe this question might be fairly trivial to someone that has more experience in such field. However, I was working on a coding ...
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Process producing concatenations of two palindromes

I'm stuck on the problem №6 from here. The handout deals with the diamond lemma, but I just can't see how it could be applied to the problem. Alice has a sheet of paper with letter A written on it ...
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How many number of non zero pairs $(a,b)$ such that $a,b$ are both palindrome numbers, and the sum of $a$ and $b$ is $A.$.

Given any number let say $N$, how many ways this can be written as the sum of the palindrome numbers. For example $1443$ there are $20$ pairs of palindrome which have sum $1443$. $(1441, 2), (1221, ...
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How is there fewer zeros in the left of 1 in this language?

From Hopcroft's Introduction to Automata Theory, Languages, and Computations, 3ed, chapter 5 p. 172: I don't quite get how can there be fewer 0's in the left of 1 ...
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When the product of a multi-digit integer and its mirror is a palindrome, can the original number have digits greater than $2$?

I am reposting a question I posted on r/mathematics. It was suggested I ask it here. My son was jotting down some multiplications for school and asked me if there were many numbers that, when ...
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How many times in the future after the 1st of January of 2019 will a palindrome date happen?

I'm trying to solve the following problem: How many times in the future (before the year 10000) will after the 1st January of 2019 happen a palindrome date. Date is in the dx-my-ymxd forum, and ...
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Is $q=37$ the only prime such that $qd$ and $q^2 d$ are base-$10$ palindromes, where $d = 9018009$?

Is $q=37$ the only prime such that $qd$ and $q^2 d$ are base-$10$ palindromes, where $d = 9018009$? Note that $$37d = 333666333$$ and $${37}^2 d = 12345654321$$ when $$d = 9018009 = {3^2}\cdot{7^2}\...
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13 votes
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Finitely many palindromes in two consecutive number bases, for fixed and distinct numbers of digits

Double palindrome: ...is a number nontrivially palindromic in two consecutive bases $b,b\pm1$ Let $d_1,d_2$ be numbers of digits in the two bases: nontrivially means $d_1,d_2\gt 1$. Let $d=\max\{...
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Introduction to Rashee numbers

Definition: A Rashee number is an integer that can form a palindrome through the iterative process of repeatedly reversing its digits and subtracting the resulting numbers. To check if the number $...
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Arbitrarily long palindromes in two consecutive number bases

Is it possible to construct an arbitrarily long double palindrome? The double palindrome of length $d$ is a number that is palindromic (digits are the same when reversed) in two consecutive ...
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Hard system in integers related to natural number representations

Update: Observing a necessary condition, and "unbalanced" variation A necessary (but not sufficient) condition for a number to be a solution to this Diophantine system (Representing "balanced" ...
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Generalization of Palindromic number

Consider a number $n > 0$ in base $b ≥ 2$, where it is written in standard notation with $k+1$ digits $a_i$ as: $${\displaystyle n=\sum _{i=0}^{k}a_{i}b^{i}}$$ with, as usual, $0 ≤ a_i < b$ ...
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How often is $N/(2N-\sigma(N))$ a palindrome (in base-$10$) if $N$ is deficient-perfect?

Let $\sigma(N)$ denote the sum of divisors of the positive integer $N$. If $(2N-\sigma(N)) \mid N$, then $N$ is said to be deficient-perfect. Note that, if $N$ is deficient-perfect, then $N/(2N-\sigma(...
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Can long numbers be "3-palindromic"?

Question Let $n$ be a number with $d\ge9$ digits when written in number base $b\ge2$. Can $n$ be $3$-palindromic? That is, does there exist $b$, such that $n$ is simultaneously a ...
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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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