I am trying to solve the following Diophantine equation with Mathematica:


It is known that there are three positive numbers that satisfy the equation above, which are on the order of $10^{81}$. Which is a big number, but not that big for Mathematica, I think.

However, neither Solve nor FindInstance can give me the solutions. The best I could do was:

    Solve[x/(y + z) + y/(x + z) + z/(x + y) == 4 && z > y > x > 1001, {x, y, z}, Integers]

which gives me a ConditionalExpression I can't work with.

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    $\begingroup$ 10^81 is still an enormous search space, and you did not even provide that bound to Solve. I would not expect a solution in a practical amount of time. $\endgroup$ – Mr.Wizard Jun 4 '17 at 13:08
  • $\begingroup$ Then if not with the above functions, how else can I solve this function with Mathematica? $\endgroup$ – Peter Dam Jun 4 '17 at 14:23
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    $\begingroup$ I think you will need to find a more intelligent (i.e. user-involved, less brute force) approach. The paper you link seems like a good place to start. Remember that Mathematica is just a tool; it cannot take the place of a mathematician. $\endgroup$ – Mr.Wizard Jun 5 '17 at 12:28
  • $\begingroup$ That seems quite reasonable. Still, Mathematica can sometimes do astounding thingy, so I thought it is worth a try. $\endgroup$ – Peter Dam Jun 6 '17 at 9:12

There is one idea. To search for the solution of the equation.


If we know any solution $(a,b,c)$ of this equation. Then it is possible to find another $(a_2, b_2, c_2)$. Make such a change.



Then the following solution can be found by the formula.







I tried this formula to simplify, but nothing happens. Maybe someone will check in Maple?

Use the given formula and first find the other solutions.


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