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Here's a question from an old examination paper:

  1. Find all $(x,y)$ in $\mathbf{Z}^{2}$ where $y$ is odd and $y^2=x^3-4$.

  2. Find all $(x,y)$ in $\mathbf{Z}^{2}$ with $y$ even and $y^2=x^3 -4$.

  3. When $(x,y)$ in $\mathbf{Z}^{2}$ where $y=2Y$ is even and $y^2=x^3-4$, show that $x=2X$ with $X, Y$ odd and that $\gcd(Y+i,Y-i) = 1+i$.

An older student who has taken the exam already told us that we should look at $\mathbf{Z}[i]$ but I don't see where to go with this information. Help is greatly appreciated.

1 Answers 1

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Write the equation as $x^3=y^2+4$. In $\mathbb Z[i]$, you have $y^2+4 = (y+2i)(y-2i)$. Any common divisor of $(y+2i)$ and $(y-2i)$ must be a divisor of $4i$. From that and the unique factorization in $\mathbb Z[i]$ you can conclude that most divisors of $y\pm 2i$ are cubes. Now ask yourself when $(a+b i)^3$ has imaginary part equal to $2$. This should get you started. I haven't checked the details, though.

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    I solved the problem. Thank you lhf.2011-11-09