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.

A palindrome is a number or any other sequence of characters which remains the same when it is reversed (read backwards).

Usually we talk about palindromic numbers (numeral palindromes) as positive integers that remain the same when their digits are reversed. As such, palindromes are dependent on the they are observed in.

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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 listed out lots of numbers that work. I haven't…
ddswsd
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Proof for all largest palindrome numbers for each power of 10

Is there a proof that shows that for any power of 10 the largest palindrome formed by the product of two numbers with equal number of digits occur in the last tenth of the given power of ten. Or another way to say it is do the two numbers start…
qw3n
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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: $$ b_1,b_2,b_3,\dots,b_n={x^{y_1},x^{y_2},x^{y_3},\dots,x^{y_n}} $$ For every combination of…
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Context free grammar: Meaning of notation ww^R

A common example in CFG is the palindrome example. These examples often contain the $\ ww^R$ notation for the string. An example from my class could be: Strings $\ ww^R$ over the alphabet $\ \Sigma = \{0,1 \} $ (a subset of palindromes over $\…
AugustM
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Probability of palindrome of length 11

What is the probability that a random bit string of length 11 is a palindrome? Anyone have an idea of how to do this? I'm not sure how to calculate the different cases.
Will eastbay
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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 example $8128$, is the sum of three…
user243301
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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 sequence which, when compacted into one number…
Stone
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Why are multiples of 11 palindromes?

One thing I noticed is that for any integer $-10
DonielF
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How to get palindrome of a number with specific base

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: 87+78= 165+ 561 = 726 + 627= 1353 + 3531 = Palindrome ! This method works…
HN Learner
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How many 8-digit palindromes are prime?

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 prime?
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Finite State Automata whose language is all palindromic strings of length 6

Give an example of a Finite State Automata $M$ such that $L(M)$ is all palindromic strings of length $6$. Solution attempt: I was thinking we could simply define a FSA where all strings of length $6$ end up in a unique halting state, and then for…
Myopic
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how many n-digit palindromes exist

I'd never seen this kind problem before, and don't know where to start. Any help is appreciated. Thank you very much! A palindrome is a number that is the same forwards and backwards. For example, $212$ and $21466412$ are palindromes. Consider an…
user321527
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recursive definition of a palindrome help

Recall that a bit string is a string using the alphabet {0, 1}. A palindrome is a string that is equal to the reversal of itself. Consider the following recursive definition of a palindrome: Basis Step: λ (the empty string) is a…
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Probability that a randomly selected bit string of length 15 is a palindrome

Question: what is the probability that a randomly selected bit string of length $15$ is a palindrome? My partial solution is that I know the $15$ length is "$10001010100001$". That's all I know. Can someone help me find and understand the solution?
guest11
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How many numbers below 2048 are binary palindromes?

I have the following Number Systems problem. "Given the numbers from $2$ to $2047$ (inclusive) represented in binary, how many are palindromes? The leading digit can't be zero." How would I go about solving this most efficiently?
hmir
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