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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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How many 5 digit palindromes are divisible by 4? [closed]

How many 5 digit palindromes are divisible by 4? what I have figured out so far: 2abc2 4abc4 6abc6 8abc8 I haven't figured out a way to actually solve the problem yet
shrineisntready's user avatar
3 votes
2 answers
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Might there be an $n^{\text{th}}$ digit of $\pi$ where the sequence becomes palindromic?

Assuming $n>1$, would it be reasonable to think there is an $n^{\text{th}}$ digit of $\pi$ where stopping there would yield a palindromic number $(3.14159...951413)$? Would it be more likely that ...
Pickelhaube808's user avatar
1 vote
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Shifted start arithmetic progression formula why it works? $a_n=a_k+(n-k)\cdot d$

Question:The number of zeros in $(10^{60}+1)^2$ is? The number of zeros in $(10^1+1)^2$ is zero. The number of zeros in $(10^2+1)^2$ is two. The number of zeros in $(10^3+1)^2$ is four. There's a ...
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Is "aa" an even or odd palindrome?

I came across a question with a solution that says this: The context-free grammar (CFG) for palindromes of even length was given: Σ = {a, b}, P → a P a | b P b | ε Give a context-free grammar (CFG) ...
just coding's user avatar
2 votes
0 answers
51 views

Finding All Four-Digit Palindrome Pairs

I am seeking assistance with a captivating mathematical problem from the Bangladesh Math Olympiad (BdMO) $2017$ Regional competition, which took place in Chattogram, Bangladesh. This intriguing ...
NISHAT TASNIM RITU's user avatar
1 vote
0 answers
69 views

Are there more Teluop-numbers?

This is not a yet known terminology , but I suggest it for Poulet-numbers with the property that they give another Poulet-number , if the decimal expansion is written down in reverse order analogue to ...
Peter's user avatar
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Are $6$ and $626$ the only palindromic numbers of the form form $5^x+1$?

I noticed that $5^1+1=6$ and $5^4+1=626$ are both palindromic numbers. Are there any palindromic numbers other than $6$ and $626$ that is form of $5^x+1$? X must be a positive integer. I think there ...
Thirdy Yabata's user avatar
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SPOJ PALIN - Why is the mirroring approach guaranteed to give you the smallest palindrome larger than N?

https://www.spoj.com/problems/PALIN/ I am trying to understand the intuition behind why this algorithm actually works. Problem statement: Given a number $N\le10^5$ generate the palindrome just after ...
ng.newbie's user avatar
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Is there a non-negative integer x that is not a palindrome but for which x == reverse_digits(x) due to an overflow?

I recently started solving some problems on LeetCode where I came across a question that asks to write a function that checks whether a non-negative integer x is a ...
Alex R's user avatar
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Why do palindromes form these arcs?

I was watching 3Blue1Brown's YouTube video "Why do prime numbers make these spirals?", and it inspired me to look for some patterns myself. So I made some Python code as follows below. How ...
Lemma's user avatar
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Shortest palindromic Egyptian representation for reciprocal integers

Consider the problem of representing the reciprocal of an integer as an Egyptian fraction where all the denominators are palindromes. i.e. write $$ \frac{1}{n} = \sum_{i} \frac{1}{a_i} $$ where $a_i$...
Peder's user avatar
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2 answers
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Palindrome with odd sum

I believe the number of palindromes that do not include 0 and sum to even $n$, such as $1+1+2+1+1=6$, is equal to $2^{n/2}$. I think this because if we consider the first half of the palindrome: if ...
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Finding large palindromic primes in the decimal expansion of pi

I'm trying to complete a coding challenge that involves finding large palindromic primes in the decimal expansion of $\pi$. I'm at the second stage of this challenge, which asks me to find the first ...
Matheus Andrade's user avatar
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Prove or disprove the lemma regarding finite iterations that leads to a palindromic number

Suppose there is a natural number $N$. Let us reverse the digits of $N$. Now let us add both the numbers e.g. if $N=98$ then add $98$ and $89$. Theorem says that if we keep doing this finite number of ...
abcdefu's user avatar
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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 $...
Math Admiral's user avatar
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4 votes
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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 ...
Rick Does Math's user avatar
1 vote
2 answers
108 views

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 ...
boaz's user avatar
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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 ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
61 views

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 ...
sfgnhsfg's user avatar
5 votes
0 answers
108 views

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 $p(a,b,k)$ ...
Peter's user avatar
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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 ...
Vepir's user avatar
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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 ...
yeeyee123's user avatar
4 votes
2 answers
198 views

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 $k \lt n$...
user110391's user avatar
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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 ...
Ev-'s user avatar
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1 answer
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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 ...
user107952's user avatar
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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 $...
Oshawott's user avatar
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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: ...
Lehs's user avatar
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0 votes
1 answer
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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 ...
Qqq's user avatar
  • 255
1 vote
1 answer
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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 ...
Francesco Boi's user avatar
1 vote
3 answers
418 views

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 ...
VLC's user avatar
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0 votes
2 answers
578 views

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
Someone's name's user avatar
1 vote
1 answer
61 views

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 ...
A. Kvåle's user avatar
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2 votes
4 answers
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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 ...
user avatar
1 vote
2 answers
238 views

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. ...
Adnan Ahmed's user avatar
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142 views

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 ...
Antonymous's user avatar
0 votes
3 answers
620 views

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 ...
calculatormaster20's user avatar
0 votes
1 answer
210 views

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 ...
Shaurya Goyal's user avatar
1 vote
0 answers
89 views

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 ...
Moko19's user avatar
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1 vote
1 answer
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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
sqrt's user avatar
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1 vote
0 answers
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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
sqrt's user avatar
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6 votes
1 answer
433 views

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 ...
Juan Moreno's user avatar
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2 votes
1 answer
2k views

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 ...
Shromi's user avatar
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2 votes
1 answer
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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 ...
Aandrea Jordan's user avatar
1 vote
0 answers
796 views

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 $...
favq's user avatar
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0 votes
0 answers
107 views

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)
ppSpp's user avatar
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1 vote
1 answer
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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 ...
zzzz's user avatar
  • 11
2 votes
2 answers
108 views

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 ...
Shane Dizzy Sukardy's user avatar
1 vote
2 answers
1k views

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 ...
DMH16's user avatar
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7 votes
1 answer
364 views

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 ...
kvardekkvar's user avatar
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6 votes
0 answers
87 views

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, ...
Lakshman's user avatar
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