Ramanujan J (2011) 25:389–403 DOI 10.1007/s11139-010-9266-x P -adic continued fractions Jordan Hirsh · Lawrence C. Washington Received: 19 March 2010 / Accepted: 22 August 2010 / Published online: 26 May 2011 © Springer Science+Business Media, LLC 2011 Abstract We study Schneider’s p-adic continued fraction algorithms. For p = 2, we give a combinatorial characterization of rational numbers that have terminating expansions. For arbitrary p, we give data showing that rationals with terminating expansions are relatively rare. Finally, we prove an analogue of Khinchin’s theorem. Keywords p-adic continued fraction · Khinchin’s theorem Mathematics Subject Classification (2000) 11A55 · 11J70 · 11K50 1 Introduction Let p be prime. Since the usual continued fraction expansions do not converge p-adically, Schneider [5, 6] defined the continued fraction expansion of a p-adic integer to be of the form b 0 + p a 1 b 1 + p a 2 b 2 + p a 3 b 3 + . . . (1) where each a i is a positive integer, b 0 ∈{0, 1, 2,...,p − 1}, and b i ∈{1, 2,...,p − 1} for i ≥ 1. It is easy to see that such an expansion converges and that each p-adic J. Hirsh 8909 Ellsworth Ct., Silver Spring, MD 20910, USA L.C. Washington ( ) Dept. of Mathematics, University of Maryland, College Park, MD 20742, USA e-mail: lcw@math.umd.edu