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Math Help - Pell's equation

  1. #1
    Super Member Showcase_22's Avatar
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    Pell's equation

    Find a pair of integers m and n satisfying:

    1). m^2-17n^2=-1

    and another pair of integers satisfying:

    2). m^2-17n^2=1

    __________________________________________________ _____

    This is the last part of the question. The first part says:

    "Work out the continued fraction expansion for \sqrt{17}".

    This comes out to be 4, \overline{8}.

    The mark scheme then uses the first two convergents to solve these equations:

    "The first two convergents are \frac{q_0}{1}, \ \frac{q_0 q_1+1}{q_1} with q_0=4 and q_1=8.

    They then use these to solve the equations.

    However, how do you know that using this irrational number will solve these equations?

    In particular, what have the first two convergents of this number got to do with finding the integers m and n?
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  2. #2
    MHF Contributor
    Jester's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    Find a pair of integers m and n satisfying:

    1). m^2-17n^2=-1

    and another pair of integers satisfying:

    2). m^2-17n^2=1

    __________________________________________________ _____
    Just the first part

    (1) (4,1)
    (2) (33,8)
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  3. #3
    Super Member Showcase_22's Avatar
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    How did you know that numbers would work?

    I can't see why the convergents to the continued fractions are also solutions to the equations.

    Why is this?
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