Hello,


Is there a solution to the following equation:


$c^n=a^{2n}+a^n b^n + b^{2n}$
where $a,b,c \in \mathbb{N}^\*$, and $n$ is an integer $\geq 2$.


The problem is due to Antoine Balan.


Thanks in advance.



Fermat-like equation $c^n=a^{2n}+a^n b^n + b^{2n}$
Writing the equation as
$$
c^n+(ab)^n = (a^n+b^n)^2,
$$
one has, via results of Darmon and Merel, and Poonen, that there
are no solutions for $n \geq 4$, provided the three terms are coprime.
If $n=2$, I think Pythagorean triply stuff shown that there are again no
solutions. In the case $n=3$, I guess one could again look at the
parametrizations for $X^3+Y^3=Z^2$, say in Cohen's GTM. I'm not sure how 
hard this case is....


The condition on coprimality might be a problem, but it's too early
in the morning for me to be sure!


