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Homework Help: Integer solutions to ax^2 + bx - cy^2 - dy = 0

  1. May 17, 2015 #1
    1. The problem statement, all variables and given/known data
    I am a hobbyist looking for solutions to ax^2 + bx - cy^2 - dy = 0 where all variables are integers and are non-zero. Is there a method of doing this effectively?

    2. Relevant equations

    3. The attempt at a solution
    I can look at the numbers produced by ax^2 + bx vs cy^2 + dy and see that they have a relationship: what I mean is if I manually find a pair of close numbers, difference = d, I find the next set of values is d+2 apart, then d + 4 and so on. So it looks as though there should be a method in algebraic terms for doing this.
  2. jcsd
  3. May 17, 2015 #2


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    Of course, the trivial solution is (0, 0). Otherwise, rewrite the equation as [itex] x(ax+b)-y(cy+d)=0[/itex].
  4. May 17, 2015 #3
    Sorry I know the variables a, b, c, and d but I don't know x, y. It looks as though I might be able to do something with modular arithmetic given that both x(ax+b) and y(cy+d) now seem to both be integer multiples in other words either x or ax+b must necessarily contain some factors in common with y and cy+d. Is there a good way to find x,y? Thanks!
  5. May 17, 2015 #4
    http://www4a.wolframalpha.com/Calculate/MSP/MSP100420ag0a9de184i4e300006aa5e486371cg88e?MSPStoreType=image/gif&s=23&w=258.&h=46. [Broken]
    You can find solutions of y in the same way. Its a bit silly though as it requires you to know all but x. You can see intuitively the set of solutions from the form Svein put it in
    Yeah you could do what your saying and then write out the set of solutions
    Last edited by a moderator: May 7, 2017
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