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How can we solve $x^3 - 2y^2 =54$ Diophantine equation in natural numbers? Modular arithmetic method is not useful maybe?

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  • $\begingroup$ Try modulo with $n$ such that $\phi(n)=$lcm$(2,3)$ $\endgroup$ – lab bhattacharjee Sep 5 '14 at 19:13
  • $\begingroup$ i am not number theorist i have no idea on elliptic curves. I try to solve elementary way. But i shared because i thought that maybe i didnt see important secret in the problem $\endgroup$ – Analysis Sep 5 '14 at 19:24
  • $\begingroup$ No secrets in the theory of elliptic curves: compare with math.stackexchange.com/questions/703137/…, math.stackexchange.com/questions/669620/…, etc. $\endgroup$ – Dietrich Burde Sep 5 '14 at 19:34
  • $\begingroup$ no i mean "trick" to solve the problem for examples how we get a quick contradiction and to show unique solution (6,9)? $\endgroup$ – Analysis Sep 5 '14 at 19:40
  • $\begingroup$ The fact that the ring of the Eisenstein integers is a UFD leads to a solution, too. $\endgroup$ – Jack D'Aurizio Sep 5 '14 at 19:40
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Since $x$ must be even, $x=2z$, we have: $$ 4z^3- y^2 = 27 \tag{1}$$ or: $$z^3 = \frac{y+3\sqrt{-3}}{2}\cdot\frac{y-3\sqrt{-3}}{2}=w\cdot\overline{w},\tag{2}$$ where $w$ lies in the ring of integers of $\mathbb{Z}[\sqrt{-3}]$, that is a UFD. This implies that, if $w$ and $\overline{w}$ are coprime, $w=\frac{y-1}{2}+3\omega$ must be a cube in the Eisenstein integers $\mathbb{Z}[\omega]$ (notice that $y$ must be odd, so $\frac{y-1}{2}\in\mathbb{Z}$), but: $$ (a+b\omega)^3 = a^3+3a^2 b\omega + 3ab^2\omega^2 +b^3 = (a^3+b^3-3ab^2)+3ab(a-b)\omega\tag{3}$$ and $3ab(a-b)=3$ has no integer solutions. The only possibility left is that $w=\sqrt{-3} u^3$ and $\overline{w}= -3v^3$ with $\gcd(u,v)=1$. This happens for $(x,y)=(6,9)$.

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  • $\begingroup$ what about (6,9)? $\endgroup$ – Analysis Sep 5 '14 at 19:22
  • $\begingroup$ There was a flaw in my original argument since I assumed that $w$ and $\overline{w}$ must be coprime in $Z[\omega]$, but there is an extra case, the one for which $w=\sqrt{-3}u^3$ and $\overline{w}=-3 v^3$ with $\gcd(u,v)=1$. Now fixed. $\endgroup$ – Jack D'Aurizio Sep 5 '14 at 19:37
  • $\begingroup$ This is the only extra case since $w-\overline{w}=3\sqrt{-3}$. $\endgroup$ – Jack D'Aurizio Sep 5 '14 at 19:38

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