ON "VERY PALINDROMIC" SEQUENCES

Title & Authors
ON "VERY PALINDROMIC" SEQUENCES
BASIC, BOJAN;

Abstract
We consider the problem of characterizing the palindromic sequences $\small{{\langle}c_{d-1},\;c_{d-2}\;,{\cdots},\;c_0\rangle}$, $\small{c_{d-1}{\neq}0}$, having the property that for any $\small{K{\in}\mathbb{N}}$ there exists a number that is a palindrome simultaneously in K different bases, with $\small{{\langle}c_{d-1},\;c_{d-2}\;,{\cdots},\;c_0\rangle}$ being its digit sequence in one of those bases. Since each number is trivially a palindrome in all bases greater than itself, we impose the restriction that only palindromes with at least two digits are taken into account. We further consider a related problem, where we count only palindromes with a fixed number of digits (that is, d). The first problem turns out not to be very hard; we show that all the palindromic sequences have the required property, even with the additional point that we can actually restrict the counted palindromes to have at least d digits. The second one is quite tougher; we show that all the palindromic sequences of length d = 3 have the required property (and the same holds for d = 2, based on some earlier results), while for larger values of d we present some arguments showing that this tendency is quite likely to change.
Keywords
palindrome;number base;heuristic;
Language
English
Cited by
References
1.
W. D. Banks and I. E. Shparlinski, Average value of the Euler function on binary palindromes, Bull. Pol. Acad. Sci. Math. 54 (2006), no. 2, 95-101.

2.
B. Basic, On d-digit palindromes in different bases: The number of bases is unbounded, Int. J. Number Theory 8 (2012), no. 6, 1387-1390.

3.
J. Besineau, Independance statistique d'ensembles lies a la fonction "somme des chiffre", Acta Arith. 20 (1972), 401-416.

4.
C. Calude and H. Jurgensen, Randomness as an invariant for number representations, in: H. Maurer, J. Karhumaki and G. Rozenberg (Eds.), Results and trends in theoretical computer science, pp. 44-66, Springer-Verlag, Berlin, 1994.

5.
J. Cilleruelo, R. Tesoro, and F. Luca, Palindromes in linear recurrence sequences, Monatsh. Math. 171 (2013), no. 3-4, 433-442.

6.
A. Cobham, On the base-dependence of sets of numbers recognizable by finite automata, Math. Systems Theory 3 (1969), 186-192.

7.
A. J. Di Scala and M. Sombra, Intrinsic palindromes, Fibonacci Quart. 42 (2004), no. 1, 76-81.

8.
S. I. El-Zanati and W. R. R. Transue, On dynamics of certain Cantor sets, J. Number Theory 36 (1990), no. 2, 246-253.

9.
E. H. Goins, Palindromes in different bases: A conjecture of J. Ernest Wilkins, Integers 9 (2009), 725-734.

10.
T. Kamae, Sum of digits to different bases and mutual singularity of their spectral measures, Osaka J. Math. 15 (1978), no. 3, 569-574.

11.
D.-H. Kim, On the joint distribution of q-additive functions in residue classes, J. Number Theory 74 (1999), no. 2, 307-336.

12.
F. Luca, Palindromes in Lucas sequences, Monatsh. Math. 138 (2003), no. 3, 209-223.

13.
F. Luca and A. Togbe, On binary palindromes of the form $10^n{\pm}1$, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 487-489.

14.
C. Mauduit, C. Pomerance, and A. Sarkozy, On the distribution in residue classes of integers with a fixed sum of digits, Ramanujan J. 9 (2005), no. 1-2, 45-62.

15.
M. Queffelec, Sur la singularite des produits de Riesz et des mesures spectrales associees a la somme des chiffres, Israel J. Math. 34 (1979), no. 4, 337-342.

16.
W. M. Schmidt, On normal numbers, Pacific J. Math. 10 (1960), 661-672.

17.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, in: Proceedings of the Washington State University Conference on Number Theory, pp. 55-67, Dept. Math., Washington State Univ., Pullman, Washington, 1971.

18.
C. L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math. 319 (1980), 63-72.