no. 9-10
Number Theory
On binary palindromes of the form 10n±1
[Sur des palindromes binaires du format 10n±1]

Présenté par : Christophe Soulé

Luca, Florian1 ; Togbé, Alain2
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, Tome 346 (2008) no. 9-10, pp. 487-489.

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

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

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.015
Luca, Florian 1 ; Togbé, Alain 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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Luca, Florian; Togbé, Alain. On binary palindromes of the form $ {10}^{n}\pm 1$. Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 487-489. doi : 10.1016/j.crma.2008.03.015. http://www.numdam.org/articles/10.1016/j.crma.2008.03.015/

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

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

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

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

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

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

Cité par 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.