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AIMS Mathematics

2022, Volume 7, Issue 2: 2237-2254. doi: 10.3934/math.2022127
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Research article

Extremal orders and races between palindromes in different bases

  • Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand
  • Received: 23 August 2021 Accepted: 07 November 2021 Published: 10 November 2021

  • MSC : 11A63, 11N37, 11N56

  • Let $ b \geq 2 $ and $ n \geq 1 $ be integers. Then $ n $ is said to be a palindrome in base $ b $ (or $ b $-adic palindrome) if the representation of $ n $ in base $ b $ reads the same backward as forward. Let $ A_b (n) $ be the number of $ b $-adic palindromes less than or equal to $ n $. In this article, we obtain extremal orders of $ A_b (n) $. We also study the comparison between the number of palindromes in different bases and prove that if $ b \neq b_{1} $, then $ A_{b} (n) - A_{b_{1}} (n) $ changes signs infinitely often as $ n \rightarrow \infty $.

    Citation: Phakhinkon Phunphayap, Prapanpong Pongsriiam. Extremal orders and races between palindromes in different bases[J]. AIMS Mathematics, 2022, 7(2): 2237-2254. doi: 10.3934/math.2022127

    Related Papers:

  • Let $ b \geq 2 $ and $ n \geq 1 $ be integers. Then $ n $ is said to be a palindrome in base $ b $ (or $ b $-adic palindrome) if the representation of $ n $ in base $ b $ reads the same backward as forward. Let $ A_b (n) $ be the number of $ b $-adic palindromes less than or equal to $ n $. In this article, we obtain extremal orders of $ A_b (n) $. We also study the comparison between the number of palindromes in different bases and prove that if $ b \neq b_{1} $, then $ A_{b} (n) - A_{b_{1}} (n) $ changes signs infinitely often as $ n \rightarrow \infty $.


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