Journal of Integer Sequences, Vol. 27 (2024), Article 24.4.8

Exact Formulas for the Number of Palindromes in Certain Arithmetic Progressions


Kritkhajohn Onphaeng
Faculty of Science and Technology
Princess of Naradhiwas University
Naratiwat 96000
Thailand

Tammatada Khemaratchatakumthorn
Department of Mathematics
Faculty of Science
Silpakorn University
Nakhon Pathom 73000
Thailand

Phakhinkon Napp Phunphayap
Department of Mathematics
Faculty of Science
Burapha University
Chonburi 20131
Thailand

Prapanpong Pongsriiam
Department of Mathematics
Faculty of Science
Silpakorn University
Nakhon Pathom 73000
Thailand
and
Graduate School of Mathematics
Nagoya University
Nagoya 464-8602
Japan

Abstract:
A positive integer n is a b-adic palindrome if the representation of n in base b reads the same backward as forward. In this article, we obtain exact formulas for the number of b-adic palindromes that are less than or equal to m and are congruent to r modulo q when b is congruent to 0 or 1 mod q. This extends Pongsriiam and Subwattanachai's result (done only for q ≤ 2), and supplements Col's theorem, which is restricted to the case that b(b2 - 1) is coprime to q.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A002113 A006995.)

Received July 1 2023; revised versions received April 2 2024; April 4 2024. Published in Journal of Integer Sequences, April 13 2024.

Return to Journal of Integer Sequences home page