Number theory, geometry and algebra - Dynamics-approx.jku.at

Similarly, one shows that
√ 1| 3 = 1+
|1 + 1|
|2 + 1|
|1 + 1| +··· = [1;1,2] (113)
|2
√
5 = [2;4] (114)
√
7 = [2;1,1,1,4] (115)
More generally, consider the fraction
x = [b;a] = b+ 1|
|a + 1|
|b + 1|
|a +...
where a,b ∈ N and a|b (say b = a.c). x is a solution of the equation
x = b+ 1|
|a + 1|
|x = (ab+1)x+b .
ax+1
(116)
Hence x 2 −bx−c = 0, i.e. x = 1 2 (b+√ b 2 +4c).
The latter are examples of periodical continued fractions i.e. those of
the form
[a 0 ;a 1 ,...,a n ,a n+1 ,...,a m ].
Proposition 29 ϑ has a periodical representation ⇔ ϑ is a quadratic irrational
number i.e. a solution of an equation
aϑ 2 +bϑ+c = 0
where a,b,c ∈ Z and d = b 2 −4ac > 0 is not a quadratic number.
Lemma 4 Let x,y ∈ R, with y > 1 and let
x =
py +r
qy +s
where p,q,r,s ∈ Z with ps−qr = ±1. Then if q > s > 0, there exists an n,
so that
p
q = p n
, resp. r q n s = p n−1
q n−1
where ( pn
q n
) is the sequence of convergents of the continued fraction representation
of x.
42