Abstract
In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.
Highlights
We are interested in the connection among the numbers with eventually periodic continued β-fraction expansion, quadratic irrational, and the corresponding Lévy constant
Our first result shows that any number with eventually periodic continued β-fraction expansion is a root of an irreducible quadratic polynomial with coefficients in Z[ β]
Continued fractions have a number of remarkable properties related to real numbers
Summary
We are interested in the connection among the numbers with eventually periodic continued β-fraction expansion, quadratic irrational, and the corresponding Lévy constant. Our first result shows that any number with eventually periodic continued β-fraction expansion is a root of an irreducible quadratic polynomial with coefficients in Z[ β]. If the continued β-fraction expansion of a positive number u is eventually periodic, there exists an irreducible quadratic polynomial ax2 + bx + c = 0, where a, b, c ∈ Z[ β] with solution u. Analog to Lagrange’s Theorem in a regular continued fraction, a natural√question is whether the converse of Theorem 1 holds We conjecture that it is not true for β = 52+1 ; that is, there exists a positive number, which is a solution of irreducible quadratic polynomial, but its continued β-fraction expansion is not eventually periodic.
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