On the regular continued fractions of real algebraic irrational numbers
https://doi.org/10.15330/cmp.18.1.258-263
Keywords:
continued fraction, irrational number, transcendental number, diophantine approximationAbstract
It is well known that an irrational number is quadratic if and only if its regular continued fraction expansion is ultimately periodic. However, no such characterization is known for other real irrational numbers. In 1949, A.Ya. Khinchin conjectured that partial denominators of the regular continued fractions of real algebraic numbers of degree higher than 2 are unbounded. In other words, if partial denominators of the regular continued fractions is bounded, then it is a quadratic number or a transcendental number.
In this paper, we observe the regular continued fractions of real algebraic numbers of degree higher than 2. More precisely, we give the minimal polynomials of the real algebraic numbers appearing in the regular continued fractions and establish their properties.
