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Every positive integer is a sum of three palindromes


Authors: Javier Cilleruelo, Florian Luca and Lewis Baxter
Journal: Math. Comp.
MSC (2010): Primary 11B13, 11A63
DOI: https://doi.org/10.1090/mcom/3221
Published electronically: August 15, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: For integer $ g\ge 5$, we prove that any positive integer can be written as a sum of three palindromes in base $ g$.


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  • [1] William D. Banks, Every natural number is the sum of forty-nine palindromes, Integers 16 (2016), Paper No. A3, 9. MR 3458332
  • [2] William D. Banks, Derrick N. Hart, and Mayumi Sakata, Almost all palindromes are composite, Math. Res. Lett. 11 (2004), no. 5-6, 853-868. MR 2106245, https://doi.org/10.4310/MRL.2004.v11.n6.a10
  • [3] William D. Banks and Igor E. Shparlinski, Average value of the Euler function on binary palindromes, Bull. Pol. Acad. Sci. Math. 54 (2006), no. 2, 95-101. MR 2266140, https://doi.org/10.4064/ba54-2-1
  • [4] William D. Banks and Igor E. Shparlinski, Prime divisors of palindromes, Period. Math. Hungar. 51 (2005), no. 1, 1-10. MR 2180629, https://doi.org/10.1007/s10998-005-0016-6
  • [5] Attila Bérczes and Volker Ziegler, On simultaneous palindromes, J. Comb. Number Theory 6 (2014), no. 1, 37-49. MR 3289196
  • [6] Javier Cilleruelo, Florian Luca, and Igor E. Shparlinski, Power values of palindromes, J. Comb. Number Theory 1 (2009), no. 2, 101-107. MR 2663607
  • [7] Javier Cilleruelo, Rafael Tesoro, and Florian Luca, Palindromes in linear recurrence sequences, Monatsh. Math. 171 (2013), no. 3-4, 433-442. MR 3090802, https://doi.org/10.1007/s00605-013-0477-2
  • [8] E. Friedman, Problem of the month (June 1999), http://www2.stetson.edu/ ~efriedma/matheoremagic/0699.html.
  • [9] M. F. Hasler, User: M. F. Hasler/ Work in progress/Sum of palindromes. From the On-Line Encyclopedia of Integer Sequences (OEIS) wiki. (Available at https://oeis.org/wiki/ User:M.F. Hasler/Work in progress/Sum of palindromes)
  • [10] Florian Luca, Palindromes in Lucas sequences, Monatsh. Math. 138 (2003), no. 3, 209-223. MR 1969517, https://doi.org/10.1007/s00605-002-0490-3
  • [11] Florian Luca and Alain Togbé, On binary palindromes of the form $ 10^n\pm1$, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 487-489. MR 2412782, https://doi.org/10.1016/j.crma.2008.03.015
  • [12] M. Sigg, On a conjecture of John Hoffman regarding sums of palindromic numbers, Preprint, 2015, arXiv 1510.07507v1.

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Additional Information

Javier Cilleruelo
Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid 28049, Madrid, España

Florian Luca
Affiliation: School of Mathematics, University of the Witwatersrand Private Bag X3, Wits 2050, South Africa
Email: florian.luca@wits.ac.za

Lewis Baxter
Affiliation: School of Applied Computing, Sheridan College, 1430 Trafalgar Road, Oakville, Ontario L6H 2L1, Canada
Email: Lewis.Baxter@SheridanCollege.ca

DOI: https://doi.org/10.1090/mcom/3221
Received by editor(s): March 2, 2016
Received by editor(s) in revised form: September 27, 2016, and June 9, 2017
Published electronically: August 15, 2017
Additional Notes: The first author was supported by MINECO project MTM2014-56350-P and by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO). The second author was supported in part by a start-up grant from Wits University and by an NRF A-rated researcher grant.
Article copyright: © Copyright 2017 American Mathematical Society

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