Abstract
Continued fractions in the field of p p -adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrangeâs Theorem). It is still not known if a p p -adic continued fraction algorithm exists that shares a similar property. In this paper we modify and improve one of Browkinâs algorithms. This algorithm is considered one of the best at the present time. Our new algorithm shows better properties of periodicity. We show for the square root of integers that if our algorithm produces a periodic expansion, then this periodic expansion will have pre-period one. It appears experimentally that our algorithm produces more periodic continued fractions for quadratic irrationals than Browkinâs algorithm. Hence, it is closer to an algorithm to which an analogue of Lagrangeâs Theorem would apply.
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