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<p>Unlike the usual simple continued fraction, Schneider’s <i>p</i>-adic continued fraction expansion of a rational number can be infinite. Hirsh and Washington conjectured that rational numbers with nonterminating expansion are more common than those with terminating expansion. We prove this conjecture and give upper and lower bounds on the number of reduced fractions, with bounded numerator and denominator, that have terminating <i>p</i>-adic continued fraction expansion. We also present examples of sets containing rationals with nonterminating expansion.</p>

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Schneider’s p-adic continued fractions of rational numbers

  • 期刊论文

  • 发表日期：

    2025年9月18日

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  • Tomislav Pejković

摘要

Unlike the usual simple continued fraction, Schneider’s p-adic continued fraction expansion of a rational number can be infinite. Hirsh and Washington conjectured that rational numbers with nonterminating expansion are more common than those with terminating expansion. We prove this conjecture and give upper and lower bounds on the number of reduced fractions, with bounded numerator and denominator, that have terminating p-adic continued fraction expansion. We also present examples of sets containing rationals with nonterminating expansion.