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Solving a Diophantine Equation

Date: 01/28/2005 at 09:01:08
From: Jonathan
Subject: Diophantine equation

Solve the following Diophantine equation:  5x + 3kx = 8k^2 - 25

We can isolate the variable x from the starting equation:

  x = (8k^2 - 25)/(3k + 5)

But what comes next?

Date: 01/28/2005 at 18:06:19
From: Doctor Vogler
Subject: Re: Diophantine equation

Hi Jonathan,

Thanks for writing to Dr. Math.  Nice question.  I love Diophantine
equations.  Your first step was exactly right.  The next thing to do
is some polynomial division on the right.  You have

      8k^2 - 25
  x = ---------,
       3k + 5

and you divide and get

       8     (-40/3)k - 25
  x = (-)k + -------------,
       3         3k + 5

and then you do it again and multiply both sides of the equation by 9
to get

  9x = 24k - 40 - ------,
                  3k + 5

at which point you find that you can also write your equation as

  (9x - 24k + 40)(3k + 5) = -25.

And now you know that 3k+5 has to be an integer factor of -25, which
means it is one of

  -25, -5, -1, 1, 5, 25.

So subtract 5 from each of those, divide each by 3, and try all of the
integers that result to solve for x.  (Note that 9x - 24k + 40 must be
the corresponding factor.)

I find two solutions.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum 

Date: 02/04/2005 at 09:50:58
From: Jonathan
Subject: Thank you (diophantine equation)

Thanks for the help and explanations!  It is very interesting indeed
and it was really easy to follow, thank you!
Associated Topics:
High School Basic Algebra
High School Number Theory

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