Continued Fraction Expansion of Algebraic Numbers

Abstract

This chapter discusses the continued fraction expansion of algebraic numbers. Every irrational x in [0, 1] determines uniquely an infinite sequence of positive integers by means of its continued fraction expansion of which ai are called the partial denominators. Conversely, every sequence of positive integers determines uniquely an irrational x in [0, 1] by (1). The conjecture does not contradict Khintchine's theorem, because the algebraic numbers form a set of Lebesgue measure zero. The conjecture was studied by numerical tests in which the continued fraction expansions of several cubic, quartic, and quintic irrationals were obtained to between 700 and 800 terms, using multi-precision integer arithmetic on the Manic I computer. The results did not support the conjecture.