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    I'm trying to follow this example (which uses the euclidean algorithm):

    Compute a greatest common divisor \delta of \alpha = 71 + 17 i and \beta=5-19i.

    \alpha=\kappa_1 \beta + \rho_1. The quotient in \mathbb{C} is

    \displaystyle \frac{\alpha}{\beta} = \frac{16+717i}{193} = 4i+\frac{16-55i}{193}

    so we take the quotient \kappa_1 to be 4i.

    I don't really understand this. Can someone explain how \kappa_1 is computed?
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    (Original post by notnek)
    I'm trying to follow this example (which uses the euclidean algorithm):

    Compute a greatest common divisor \delta of \alpha = 71 + 17 i and \beta=5-19i.

    \alpha=\kappa_1 \beta + \rho_1. The quotient in \mathbb{C} is

    \displaystyle \frac{\alpha}{\beta} = \frac{16+717i}{193} = 4i+\frac{16-55i}{193}

    so we take the quotient \kappa_1 to be 4i.

    I don't really understand this. Can someone explain how \kappa_1 is computed?
    My guess is that in this step \displaystyle \frac{\alpha}{\beta} = \frac{16+717i}{193} = 4i+\frac{16-55i}{193} they're seperating k1 and p, getting \displaystyle \frac{\alpha}{\beta}=\kappa_1 +\displaystyle \frac {\rho_1}{\beta}

    So, k is the non fractional part, so 4i.
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    (Original post by Slumpy)
    My guess is that in this step \displaystyle \frac{\alpha}{\beta} = \frac{16+717i}{193} = 4i+\frac{16-55i}{193} they're seperating k1 and p, getting \displaystyle \frac{\alpha}{\beta}=\kappa_1 +\displaystyle \frac {\rho_1}{\beta}

    So, k is the non fractional part, so 4i.
    But \displaystyle \frac{16+717i}{193}=3i+\frac{16+  138i}{193}

    Why not 3i?

    I think this may have something to do with a lattice of squares which can compare two gaussian integers but I'm not sure.
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    You want the "remainder" to be as small as possible, and (16+138i) is a lot larger than (16-55i).
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    (Original post by DFranklin)
    You want the "remainder" to be as small as possible, and (16+138i) is a lot larger than (16-55i).
    OK but if I only have \frac{\alpha}{\beta}, how do I find \kappa_1 because it could be anything. In another example I have, it's (1+i).
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    http://fermatslasttheorem.blogspot.c...-gaussian.html
 
 
 
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