Periodic representations for quadratic irrationalities in the field of p-adic numbers

Continued fractions have been widely studied in the field of p-adic numbers Qp, 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 standard, for any quadratic irrational via p-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 R and Qp. Moreover given two primes p1 and p2, using the Binomial transform, we are also able to pass from approximations in Qp1 to approximations in Qp2 for a given quadratic irrational. Then, we focus on a specific p–adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in [6]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.