Antipalindromic numbers

Authors

  • Ľubomíra Dvořáková Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Department of Mathematics, Trojanova 13, 120 00 Praha 2, Czech Republic
  • Stanislav Kruml Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Department of Mathematics, Trojanova 13, 120 00 Praha 2, Czech Republic
  • David Ryzák Charles University, Faculty of Mathematics and Physics, Ke Karlovu 2027, 121 16 Praha 2, Czech Republic

DOI:

https://doi.org/10.14311/AP.2021.61.0428

Keywords:

Expansion in a base, palindromes, antipalindromes, palindromic numbers, antipalindromic numbers.

Abstract

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.

 

Comment after publishing (November 2021)

A reader has found numbers 43490045690,  45741687530,  51300998210,  56554588825 and 667298874550 that are antipalindromic in bases 6 and 8. Therefore 6 and 8 are not suitable candidates in the open problem (4.).

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References

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.

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Published

2021-06-30

How to Cite

Dvořáková, Ľubomíra ., Kruml, S., & Ryzák, D. (2021). Antipalindromic numbers. Acta Polytechnica, 61(3), 428–434. https://doi.org/10.14311/AP.2021.61.0428

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Articles