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Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question.

What can you add to the following list of applications?

1) Calculation and approximation of quadratic irrational numbers. calculation of corresponding covex hulls.

2) Pell equation and calculation of fundamental units in quadratic fields.

3) Reduction of quadratic forms. Calculation of class numbers of imaginary quadratic field.

4) Legendre's factorization method.

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What is the question? –  André Henriques Dec 20 '10 at 7:42
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2 Answers

There is a pleasant connection between (among?) Chebyshev polynomials, the Pell equation and continued fractions, the latter two being understood to take place in real quadratic function fields rather than the "classical" case of real quadratic number fields.

It's been a while since I saw the details, but upon recent cursory inspection it seems that a treatment of this can be found in Section 3.4 of Edward J. Barbeau's book Pell's Equation.

Anyway, this circle of ideas has made me think that continued fractions in the function field case should possibly get a more prominent treatment in introductory number theory texts. (This is of course pretty antithetical to the strict "no continued fractions" policy in my own number theory notes. Insert the standard Whitman quote about self-contradiction and multitudes here.)

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As I understand, Section 3.3 of Edward J. Barbeau's book is also important for your answer. –  Alexey Ustinov Dec 20 '10 at 14:20
    
@Alexey: Thanks, I'm sure you're right. By the way, since it is CW, I invite anyone to flesh out what is currently a pretty vague response. –  Pete L. Clark Dec 20 '10 at 19:22
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The conjugacy problem in $SL(2,Z)$. For matrices $M \in GL(2,Z)$ having trace of absolute value $>2$, the slope of its expanding eigenvector has an eventually periodic continued fraction expansion (it is a quadratic irrational), and the primitive period loop is a conjugacy invariant in $SL(2,Z)$. Throw in the absolute value of the trace itself and you have a complete conjugacy invariant in $SL(2,Z)$.

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