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<p>Continued fractions have been generalized over the field of <i>p</i>-adic numbers, where it is still not known an analogue of the famous Lagrange’s Theorem. In general, the periodicity of <i>p</i>-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic <i>p</i>-adic continued fractions and convergence to real quadratic irrationals. In particular, in the first part we prove that convergence in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> is a necessary condition for the periodicity of the continued fraction of a quadratic irrational in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {Q}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Q</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>. 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 <i>p</i>-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the <i>p</i>-adic digits of a quadratic irrational, which holds for almost all <i>p</i>-adic numbers.</p>

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Real convergence and periodicity of p-adic continued fractions

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  • 发表日期：

    2025年11月20日

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  • Giuliano Romeo

摘要

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 convergence to real quadratic irrationals. In particular, in the first part we prove that convergence in \(\mathbb {R}\) R is a necessary condition for the periodicity of the continued fraction of a quadratic irrational in \(\mathbb {Q}_p\) 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, which holds for almost all p-adic numbers.