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With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$
I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into $x(x^2-y^2)=N-1$ and then tried to link it to Pell's Equation but so far I've got nothing, I don't even understand how Pell's Equation is solved (even though I know the method, I cannot grasp the intuition or reasoning behind it), I would appreciate any help.

GuPe
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    there is no link to Pell or much of anything else. The reasonable procedure is to find all the divisors of $N-1,$ also the negatives of these. Then all three of $x,x-y,x+y$ must be in the list. Where did you get this problem??????? – Will Jagy Jul 20 '15 at 00:43
  • @WillJagy, is there a method that does not require brute force? I was trying to find the link to Pell because the Pell Equation does not require much brute force after finding the approximation via continued fractions. I was hoping this expression had similar solutions. Is the origin of the problem important? If so I can edit the post to bring in more detail regarding its genesis. But generally I thought such details were superfluous :) – GuPe Jul 20 '15 at 01:11
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    I think you had better tell the whole story of this, and give some information on your own background as relates to mathematics. The one you gave has a wonderful algorithmic answer, just not closed form. Oh, the whole point of Pell is indefinite quadratic forms that *cannot* be factored and the solution set is infinite, you have $x(x+y)(x-y)$ and finitely many solutions. – Will Jagy Jul 20 '15 at 01:34
  • @willjagy alright, but does it really have no closed form solution? – GuPe Jul 21 '15 at 13:04
  • Compare $xy+1=N$, which you would take to $xy=N-1$ and factor $N-1$. Is that a closed form? In this case you have three factors and need an arithmetic progression between them. – Ross Millikan Jul 24 '15 at 15:06
  • If the origin of the problem is a textbook that says "find a closed form solution", this is much different than "I was playing around with this equation". It can give us some indication of what to expect out of the problem. You also seem a bit surprised at no closed form solution - in general when a solution to a problem depends on the prime factorization of a number, there will most likely be no closed form solution. – RghtHndSd Jul 24 '15 at 15:18
  • I agree with comments, the solution(s) depend on factorization of $N-1$, and you have $x(x-y)(x+y)$ to fit this, which has two variables, and you don't even know the prime factorization of $N-1$. (There are no known closed form formulas for prime factorizations, just like there are no known elementary generative formulas that give all primes). It may even turn out that finding all solutions for $x,y$ given $N$ is NP-hard on a regular (non-quantum) computer, although it's so close to simple factorization that actually proving hardness probably can't be done at the moment. – user2566092 Jul 28 '15 at 16:43

1 Answers1

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You could always solve for $y$:$$y=\pm\sqrt{\frac{N-1-x^3}{-x}}$$ From here, you could use brute force to find some integer $x$ value that returns integer $y$ value.

You could always do it backwards as well, solving for $x$.

This is more difficult, as you are trying to solve a cubic equation. However, there is a cubic formula with much rigorous derivation behind it.

It can be found here.

Sadly, you would still have to use brute force methods. But the cube roots and such will make finding a $y$ value that will work much narrower.

Simply Beautiful Art
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