The minimal continued fraction of m (where 0<m∈Z is not a square) is connected with the corresponding simple continued fraction, from which it can be written out.In this paper, it is shown that the minimal conti...The minimal continued fraction of m (where 0<m∈Z is not a square) is connected with the corresponding simple continued fraction, from which it can be written out.In this paper, it is shown that the minimal continued fraction is periodic, its period is shorter than twice of the period of the corresponding simple continued fraction, its absolute\|period is not greater than the period of the corresponding simple continued fraction.Several properties of the minimal continued fraction are also obtained.展开更多
Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 ...Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].展开更多
基金Supported by the Science Foundation of Tsinghua Uni-versity
文摘The minimal continued fraction of m (where 0<m∈Z is not a square) is connected with the corresponding simple continued fraction, from which it can be written out.In this paper, it is shown that the minimal continued fraction is periodic, its period is shorter than twice of the period of the corresponding simple continued fraction, its absolute\|period is not greater than the period of the corresponding simple continued fraction.Several properties of the minimal continued fraction are also obtained.
文摘Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].
