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We want to find set of all values that satisfy the following equation: $(a+ky)(a-ky)=gx$ All values are assumed to be nonzero integers. How does one set $x$ so that $a$ is not multiples of $y$ while there exist more than one unique solution set of other values? In this case, $a$, $k$ and $g$ are allowed to change, while $y$ must be fixed. Also, how does one compute the number of possible combinations of solutions? How does the value of $x$ relate to the number of possible solutions?

With the same constraint, what happens if $g$ is fixed to 1? And with the same constraint, what happens if we allow for $k$ of each solution to be a fraction of the form $1/q$ where $q$ is some factor of $y$ and $q$ of each solution does not have any prime factor that is a prime factor of $q$ of other solutions?

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    It isn't easy to see from the question which of $a,k,g,x,y$ are considered fixed and which are to be "solved for". By saying "$a,k,g$ are allowed to change, it looks like you are viewing $x,y$ as fixed, and then searching for $a,k,g$. Is that it?2012-12-29
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    Yes $x$ and $y$ is set to be fixed. But the question asks which $x$ and $y$ allows the equation with constraints to be solved. And other questions.2012-12-29
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    @coffeemath is this a difficult question?2012-12-29
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    Given any $x,y$ define $a=x+y$ so that $a-y=x$, then define $a+y=g$, and so with $k=1$ you have $(a+ky)(a-ky)=gx$.2012-12-30
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    My above comment is one solution to the first version. To find *all* solutions might be difficult.2012-12-30

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