Warm-up Diophantine Equation.

freemind

Warm-up Diophantine Equation.

**Posted:** Mon Mar 03, 2008 5:51 pm

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The fate of equilibrium is to end the eternity...

Umut Varolgunes

$x^3 + y^3 - 3xy + 1 = (x + y + 1)(x^2 + y^2 + 1 - x - y - xy) = p$
$x = y = 1$ doesn't give any solution so $x + y > 0$ . hence $x^2 + y^2 + 1 - x - y - xy = 1$
if $x$ and $y$ are bigger than 1; $\frac {x^2 + y^2}{2} > = x + y}$ and $\frac {x^2 + y^2}{2} > = xy$ so $x^2 + y^2 > = xy + x + y$ and in the equality case $x = y = 2$ gives $p = 5$
if $x$ or $y$ is $1$ say $x = 1$ . $y^2 - 2y = 0$ so $y = 2$ or $y = 0$ . checking yields only $y = 0$ is a solution.
so the solutions are $(2,2,5), (0,1,2), (1,0,2)$