Re:Golden powers are nearly integers
by winstonsmith on Thursday March 23, 2017 @09:23AM
(#13939470)
>To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0.
Noting that {F(n+1)/F(n)} is the sequence of convergents of the continued fraction expansion of φ, your stronger condition follows from Theorem 5 at https://en.wikipedia.org/wiki/Continued_fraction .
To show what we want, we must prove the stronger condition that F(n)φ - F(n+1) → 0. Merely knowing that F(n+1)/F(n) → φ only shows that the difference grows more slowly than F(n).