A noble number ν is defined as an irrational number having a continued fraction that becomes an infinite sequence of 1s at some point, ν congruent [0, a_1, a_2, ..., a_n, 1^_]. The prototype is the inverse of the golden ratio ϕ^(-1), whose continued fraction is composed entirely of 1s (except for the a_0 term), [0, 1^_]. Any noble number can be written as ν = (A_n + ϕ^(-1) A_(n - 1))/(B_n + ϕ^(-1) B_(n + 1)), where A_k and B_k are the numerator and denominator of the kth convergent of [0, a_1, a_2, ..., a_n].

near noble number | periodic continued fraction