# 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
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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 ...
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) ...
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### 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 ...
1 vote
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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $a_i$...
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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 ...
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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 ...
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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, ...