Solving a Diophantine EquationDate: 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 25 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 http://mathforum.org/dr.math/ 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! |
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