The Continued Fraction Expansion of An Algebraic Power Series Satisfying A Quartic Equation

Some time ago Mills and Robbins (1986, J. Number Theory23, No. 3, 388-404) conjectured a simple closed form for the continued fraction expansion of the power series solution ƒ = a1x−1 + a2x−2 + · · · to the equation ƒ4 + ƒ2 − xƒ + 1 = 0 when the base field is GF(3). In this paper we prove this conjecture. Mills and Robbins also conjectured some properties of the continued fraction expansion when the base field was GF(13). We extend this conjecture by giving the continued fraction expansion in the GF(13) case explicitly.