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Is there an analogue of the Simple Continued Fraction in p-adic number fields? Is it useful and does it have relations to best rational approximation in the p-adic sense? In the analytic case there is the P-fraction and the associated Pade'-approximants. The Pade' approximants are best rational approximants in the x-adic metric.

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  • $\begingroup$ Have you tried developing the theory along the lines it follows in the reals, to see how far you can get? $\endgroup$ May 24, 2016 at 22:57
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    $\begingroup$ Do you have access to MathSciNet (Math Reviews online)? If so, type in "p-adic" and "continued fraction*" and you'll get 100 references, such as MR2986927 Kojima, Michitaka; Continued fractions in p-adic numbers. Algebraic number theory and related topics 2010, 239–254, RIMS Kôkyûroku Bessatsu, B32, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. And MR2997750 Hančl, J.; Jaššová, A.; Lertchoosakul, P.; Nair, R. On the metric theory of p-adic continued fractions. Indag. Math. (N.S.) 24 (2013), no. 1, 42–56. $\endgroup$ May 24, 2016 at 23:01
  • $\begingroup$ Gerry, thanks for your suggestions. I am very isolated at this time and don't have access to MathSciNet. If you have any links to papers I would appreciate it. Thanks again. $\endgroup$ May 25, 2016 at 3:15
  • $\begingroup$ I typed $$\rm p-adic\ continued\ fraction$$ into Google and this came out on top: math.arizona.edu/~ura-reports/061/Moore.Matthew/Final.pdf You might try doing the same, see what turns up. $\endgroup$ May 25, 2016 at 5:29
  • $\begingroup$ Gerry, thanks again. I have not learned the power of Google. There is much in the public domain on this topic. I'm feeling a little the dunce today! $\endgroup$ May 25, 2016 at 18:22

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