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Show that the following equation has no integer solutions:

$x^3+3x^2+2x=z^3-4z+4.$

No idea where to start because it has no $y$ functions.

Also I need to find the integer solutions to $y^2+x^2=9-z^2$.

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  • $\begingroup$ For your second equation note that $x^2+y^2+z^2=9$, so $-3\le x,y,z\le 3$. $\endgroup$ – Xam Dec 1 '16 at 15:22
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    $\begingroup$ For the second question, you should really make a second post. However, there are only a few possibilities for $x^2+y^2+z^2=9$ since $|x|,|y|,|z|\le3$. $\endgroup$ – robjohn Dec 1 '16 at 15:22
  • $\begingroup$ If you change all the $z$s to $y$ in the first equation then you have $y$ functions but no $z$ functions. Is that better? $\endgroup$ – David K Dec 1 '16 at 15:22
  • $\begingroup$ If i change z to y, factorise both side to x(x+1)(x+2)-y(y-2)(y+2)=z^2 , where z=2. Where do i go from there, or am i going in the wrong direction? $\endgroup$ – K_uddin Dec 1 '16 at 15:44
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$x^3+3x^2+2x=x(x+1)(x+2)$ divizible by $3$. Then applying mod 3 we get: $0=1$ because of $a^p = a \mod p$ when $p$ prime.

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  • $\begingroup$ pretty similar to my hint $\endgroup$ – robjohn Dec 1 '16 at 15:54
  • $\begingroup$ @robjohn Yes, Fermat LT is the key here $\endgroup$ – user261263 Dec 1 '16 at 15:56
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Hint: Look at the equation mod $3$ using Fermat's Little Theorem.

More hints: $$ \begin{align} x^3&\equiv x&\pmod3\\ 3x^2&\equiv0&\pmod3\\ 2x&\equiv-x&\pmod3\\ \\ z^3&\equiv z&\pmod3\\ 4z&\equiv z&\pmod3\\ 4&\equiv1&\pmod3 \end{align} $$

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