Math. Comput. | 2021
Periodic representations for quadratic irrationals in the field of p-adic numbers
Abstract
<p>Continued fractions have been widely studied in the field of <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= p >\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding= application/x-tex >p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-adic numbers <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= double-struck upper Q Subscript p >\n <mml:semantics>\n <mml:msub>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >Q</mml:mi>\n </mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >\\mathbb Q_p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call <italic>standard</italic>, for any quadratic irrational via <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= p >\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding= application/x-tex >p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= double-struck upper R >\n <mml:semantics>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >\\mathbb R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= double-struck upper Q Subscript p >\n <mml:semantics>\n <mml:msub>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >Q</mml:mi>\n </mml:mrow>\n <mml:mi>p</mml:mi>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >\\mathbb Q_p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Moreover given two primes <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= p 1 >\n <mml:semantics>\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >p_1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= p 2 >\n <mml:semantics>\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >p_2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, using the Binomial transform, we are also able to pass from approximations in <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= double-struck upper Q Subscript p 1 >\n <mml:semantics>\n <mml:msub>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >Q</mml:mi>\n </mml:mrow>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >\\mathbb {Q}_{p_1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to approximations in <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= double-struck upper Q Subscript p 2 >\n <mml:semantics>\n <mml:msub>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >Q</mml:mi>\n </mml:mrow>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >\\mathbb {Q}_{p_2}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for a given quadratic irrational. Then, we focus on a specific <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= p >\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding= application/x-tex >p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>–adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp.\xa01281–1292]. Finally, we study the periodicity of this algorithm showing when it produces <italic>standard</italic> representations for quadratic irrationals.</p>