On binary palindromes of the form $ {10}^{n}\pm 1$

Florian Luca; Alain Togbé

doi:10.1016/j.crma.2008.03.015

Number Theory

On binary palindromes of the form

10^{n} \pm 1

Presented by: Christophe Soulé

Florian Luca ¹; Alain Togbé ²

¹ Instituto de Matemáticas UNAM, Campus Morelia Apartado Postal 27-3 (Xangari), C.P. 58089, Morelia, Michoacán, Mexico
² Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville, IN 46391, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 487-489.

Abstract
Résumé

In this Note, we find all positive integers n such that $10^{n} \pm 1$ is a binary palindome. Our proof uses lower bounds for linear forms in logarithms of rational numbers.

Dans cette Note, nous trouvons tous les entiers positifs n tels que $10^{n} \pm 1$ soit un palindrome binaire. Notre démontration utilise les minorations de formes linéaires en logarithmes de nombres rationnels.

Received: 2007-08-30
Accepted: 2008-03-11
Published online: 2008-04-14

DOI: 10.1016/j.crma.2008.03.015

Author's affiliations:

Florian Luca ¹; Alain Togbé ²

¹ Instituto de Matemáticas UNAM, Campus Morelia Apartado Postal 27-3 (Xangari), C.P. 58089, Morelia, Michoacán, Mexico
² Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville, IN 46391, USA

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     title = {On binary palindromes of the form $ {10}^{n}\pm 1$},
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Florian Luca; Alain Togbé. On binary palindromes of the form $ {10}^{n}\pm 1$. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 487-489. doi : 10.1016/j.crma.2008.03.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.015/

References
Cited by

[1] C. Ashbacher More on palindromic squares, J. Recreational Math., Volume 22 (1990), pp. 133-135

[2] W.D. Banks; D.N. Hart; M. Sakata Almost all palindromes are composite, Math. Res. Lett., Volume 11 (2004), pp. 853-868

[3] W.D. Banks; I. Shparlinski Prime divisors of palindromes, Period. Math. Hungar., Volume 51 (2005), pp. 1-10

[4] M. Keith Classification and enumeration of palindromic squares, J. Recreational Math., Volume 22 (1990), pp. 124-132

[5] M. Laurent; M. Mignotte; Yu. Nesterenko Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory, Volume 55 (1995), pp. 285-321

[6] F. Luca Palindromes in Lucas sequences, Monatsh. Math., Volume 138 (2003), pp. 209-223

Cited by Sources:

^⁎ Work by the first author was done in the Summer of 2007 when he visited the School of Mathematics of the Tata Institute in Mumbai, India. He thanks the host institution for its hospitality and the Third World Academy of Sciences for support. The second author was partially supported by Purdue University North Central.

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