Keywords:Expansion in a base, palindromes, antipalindromes, palindromic numbers, antipalindromic numbers.
Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance, kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer base is a palindrome. The following problems are studied: palindromic primes, palindromic squares and higher powers, multi-base palindromic numbers, etc. In this paper, we define and study antipalindromic numbers: positive integers whose expansion in a certain integer base is an antipalindrome. We present new results concerning divisibility and antipalindromic primes, antipalindromic squares and higher powers, and multi-base antipalindromic numbers. We provide a user-friendly application for all studied questions.
J. Joyce. Ulysses. First edition. Shakespeare and Company, 12, Rue de l’Odéon, Paris, 1922.
N. J. A. Sloane. The on-line encyclopedia of integer sequences. https://oeis.org.
A. Tripathi. Characterization and enumeration of palindromic numbers whose squares are also palindromic. Rocky Mountain J Math 50(3):1115–1124, 2020. https://doi.org/10.1216/rmj.2020.50.1115.
G. J. Simmons. On palindromic squares of non-palindromic numbers. J Recreational Math 5(1):11–19, 1972.
G. J. Simmons. Palindromic powers. J Recreational Math 3:93–98, 1970.
W. D. Banks, D. N. Hart, M. Sakata. Almost all palindromes are composite. Math Res Lett 11(5-6):853–868, 2004. https://doi.org/10.4310/MRL.2004.v11.n6.a10.
J. Cilleruelo, F. Luca, L. Baxter. Every positive integer is a sum of three palindromes. Math Comp 87(314):3023– 3055, 2018. https://doi.org/10.1090/mcom/3221.
A. Rajasekaran, J. Shallit, T. Smith. Additive number theory via automata theory. Theory Comput Syst 64(3):542–567, 2020. https://doi.org/10.1007/s00224-019-09929-9.
S. Kruml. Antipalindromic numbers (application). [2020-08-10], https: //github.com/Kruml3/Antipalindromic-numbers/.
J. Cilleruelo, F. Luca, I. E. Shparlinski. Power values of palindromes. J Comb Number Theory 1(2):101–107, 2009.
B. Bašic. On d-digit palindromes in different bases: the number of bases is unbounded. Int J Number Theory 8(6):1387–1390, 2012. https://doi.org/10.1142/S1793042112500819.
B. Bašic. On “very palindromic” sequences. J Korean Math Soc 52(4):765–780, 2015. https://doi.org/10.4134/JKMS.2015.52.4.765.
A. Bérczes, V. Ziegler. On simultaneous palindromes. J Comb Number Theory 6(1):37–49, 2014.
S. Kruml. Antipalindromic numbers. bachelor thesis, Czech Technical University in Prague, 2020. Available on request.
Copyright (c) 2021 Ľubomíra Dvořáková, Stanislav Kruml, David Ryzák
This work is licensed under a Creative Commons Attribution 4.0 International License.