For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.
References
[1]
Delahaye, J.P. (2008) Déconcertantes Conjectures, Pour la Science. https://www.pourlascience.fr/sr/logique-calcul/deconcertantes-conjectures-3819.ph
[2]
Kamath, K.K. and Shankar, B.R. (2010) One Time Pad via Lychrel Number and Elliptic Curves. International Journal of Computation and Applied Mathematics, 5, 157-161.
[3]
Kolpakov, U. and Kulcherov, G. (2009) Searching for Gapped Palindromes. Journal of Theoretical Computer Science, 410, 5365-5373. https://doi.org/10.1016/j.tcs.2009.09.013
[4]
Trigg, C. (1967) Palyindromes by Addition. Mathematic Magazine, 40, 26-28. https://doi.org/10.2307/2689178
[5]
Wilson, D. (2008) From Mathworld—A Wolfram Web Resource. https://mathworld.wolfram.com/196-Algorithm.html
[6]
Irvin, T. (1995) About Two Months of Computing, or, An Addendum to Mr. Walker’s Three Years of Computing. http://www.fourmilab.ch/documents/threeyears/two_months_more.html
[7]
VanLandingham, W. (2006) 196 and Other Lychrel Numbers. http://www.p196.org/
[8]
Abida, L., Sghiar, A. and Sghiar, M. (2017) Brisure de Symétrie et Nombres de Lychrel. IOSR Journal of Computer Engineering, 19, 91-94.
[9]
Hu, X.S., Clair, D., Labesse-Jied, F. and Fogli, M. (2014) A Probabilistic Approach for Studying the Reliability of Cementless Hip Prostheses in the Presence of Mechanical Uncertainties. World Journal of Mechanics, 4, 12-23. https://doi.org/10.4236/wjm.2014.41002
[10]
Houston, L.M. (2024) Non-Recursive Base Conversion Using a Deterministic Markov Process. Journal of Applied Mathematics and Physics, 12, 2112-2118. https://doi.org/10.4236/jamp.2024.126129
[11]
dCode07 (2007) Nombre de Lychrel. https://www.dcode.fr/nombre-Lychrel
