Let $f(x,y)=0$ be irreducible elliptic curve over the rationals.

Are there $f$ for which:

Both $x,y$ are arbitrary large powers infinitely often,
i.e. infinitely many rational points $(u^k,v^m)$ with
both $k,m$ arbitrary large?

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For $x,y$ squares (asked in previous revision) this is possible.
Take $f(x,y)=x^{6} - 2 x^{3} y^{3} + y^{6} - 72 x^{3} - 72 y^{3} + 1296$.

$f(x^2,y^2)$ is divisible by $x^3 + y^3 - 6$ which is genus $1$
of positive rank.




For large $k,m$ the n-conjecture implies the number of monomials
can't be too small.