Periodic Continued Fractions

Proposition Suppose that α has an eventually periodic continued fraction expansion. Then α is a quadratic irrational.

Proof. We first show this when α has a periodic continued fraction expansion. We then have a d such that

α=a0+1a1+1a2++1ad1+1α

Since α0,α1,,αd1 are all integers

α=xα+yzα+wzα2+(wx)αy=0

Since α is irrational, z0. Thus, α is a quadratic irrational.

If α=[α0,α1,,αm,αm+1,,αm+d1,αm+d,], then

β=111αα0α1αm2αm1

Clearly β has a periodic continued fraction expansion. So it is quadratic irrational.

Note,

β=xα+yzα+wa(xα+yzα+w)2+b(xα+yzα+w)+c=0aα2+bα+c=0

Since α is irrational, a0. Thus, α is a quadratic irrational.

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