Periodic representations for quadratic irrationals in the field of

Continued fractions have been widely studied in the field of p p -adic numbers alttext="double-struck upper Q Subscript p"> Q encoding="application/x-tex">\mathbb Q_p , but currently there is no algorithm replicating all good properties that continued over real regarding, particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which will call standard, for any quadratic irrational via fractions, even if it not obtained by specific algorithm. This representation provides simultaneous rational approximations both R"> mathvariant="double-struck">R R . Moreover given two primes alttext="p 1"> 1 encoding="application/x-tex">p_1 2"> 2 encoding="application/x-tex">p_2 using Binomial transform, are also able to pass from p {Q}_{p_1} {Q}_{p_2} irrational. Then, focus on –adic fraction proving stops finite number steps when processes numbers, solving problem left open paper Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, study periodicity showing produces standard representations irrationals.