Back to Search
Start Over
Real convergence and periodicity of $p$-adic continued fractions
- Publication Year :
- 2024
-
Abstract
- Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic $p$--adic continued fractions and the convergence to real quadratic irrationals. In particular, in the first part we prove that the convergence in $\mathbb{R}$ is a necessary condition for the periodicity of the continued fractions of a quadratic irrational in $\mathbb{Q}_p$. Moreover, we leave several conjectures on the converse, supported by experimental computations. In the second part of the paper, we exploit these results to develop a probabilistic argument for the non-periodicity of Browkin's $p$-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the $p$-adic digits of a quadratic irrational, that holds for almost all $p$-adic numbers.
- Subjects :
- Mathematics - Number Theory
11J70, 11D88, 11Y65, 12J25
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2410.09215
- Document Type :
- Working Paper