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Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots p_k^{z_k}$$ in $(x,y,z_1,\ldots,z_k) \in \mathbb{Z}^2 \times \mathbb{Z}_{\geq 0}^{k}$ with $\gcd(x,y)=1$?

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  • $\begingroup$ Yes: $x=1$ and everything else is 0. $\endgroup$
    – eric
    Commented Nov 12, 2014 at 19:49
  • $\begingroup$ @eric : Naturally, I meant to find all the solutions. Not just one of them. The question has been edited to clarify this. $\endgroup$
    – Jason Sawyer
    Commented Nov 12, 2014 at 19:54
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    $\begingroup$ If the left-hand side is irreducible over the rationals and you go to the quadratic field where it factors, then you are looking at the $S$-units of that field where $S=\{p_1,\ldots,p_k\}$ and the generalization of the Dirichlet unit theorem to $S$-units describes all the solutions. $\endgroup$
    – Felipe Voloch
    Commented Nov 12, 2014 at 20:25
  • $\begingroup$ It seems difficult to imagine an algorithm which could enumerate such a possibly infinite set. Certainly there is a very simple algorthim which will find all solutions given an infinite amount of time. $\endgroup$
    – Daniel Loughran
    Commented Nov 12, 2014 at 20:46
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    $\begingroup$ Why do you need linear forms in logs? Also if $b=0,c=-2$, there are infinitely many solutions with $z_i=0$, for example. $\endgroup$
    – Felipe Voloch
    Commented Nov 12, 2014 at 20:57

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