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$\textbf{How to attempt proving that $4x^3-3z^2=1$ has no positive integer solutions?} $

I want to show there is no positive integer solution to this equation other than $(1,1)$. How to do that? Is there a procedure or theorem that could be used here?

The above equation could be transformed to a Mordell Elliptic curve:

$$ t^3-s^2=432 $$ $$ s=36x,\ t=12z $$ Hence $(s,t)=(36,12)$ is a solution. Also due to Mordell, this has been shown that this type of curve can only have finitely many integer points. How to show there is only one in this case?

My reasoning used Hall's conjecture. But could it be done in any other way?

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    $\begingroup$ What about $(1,-1)$? $\endgroup$
    – Jason Starr
    Commented Mar 27, 2018 at 9:26
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    $\begingroup$ Oh. I forgot to mention positive. I have edited the question. $\endgroup$
    – diffusiondiver11
    Commented Mar 27, 2018 at 9:28
  • $\begingroup$ Related: mathoverflow.net/a/7979/116794 $\endgroup$
    – Simon L Rydin Myerson
    Commented Mar 27, 2018 at 9:30
  • $\begingroup$ More related: mathoverflow.net/questions/296267/… $\endgroup$
    – Gerry Myerson
    Commented Mar 27, 2018 at 10:38
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    $\begingroup$ Unfortunately, the usual way to prove $t^3-s^2=432$ has only the solutions you know about is by deducing it from the nonexistence of nontrivial solutions to $x^3+y^3=z^3$. $\endgroup$
    – Gerry Myerson
    Commented Mar 27, 2018 at 11:19

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