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

Is there $f$ for which:

1 Both $x,y$ are squares infinitely often, i.e. infinitely many rational points of the form $(u^2,v^2)$?

2 Both $x,y$ are (arbitrary) large powers infinitely often, i.e. infinitely many rational points $(u^k,v^m)$ for some $k,m \ge 2$ (the larger the better)?

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

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?

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.

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

Is there $f$ for which:

1 Both $x,y$ are squares infinitely often, i.e. $x=u^2,y=v^2$?

2 Both $x,y$ are (arbitrary) large powers infinitely often: $x=u^k,y=v^m$ for some $k,m \ge 2$ (the larger the better)?

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

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

Is there $f$ for which:

1 Both $x,y$ are squares infinitely often, i.e. infinitely many rational points of the form $(u^2,v^2)$?

2 Both $x,y$ are (arbitrary) large powers infinitely often, i.e. infinitely many rational points $(u^k,v^m)$ for some $k,m \ge 2$ (the larger the better)?

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

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

Is it possible:

1 Both $x,y$ to be squares infinitely often?

2 Both $x,y$ to be (arbitrary) large powers, not necessarily the same power?

For Weierstrass model, $x$ square infinitely often is possible, though I believe abc implies both can't be squares infinitely often.

For larger powers the n-conjecture implies the number of monomials can't be too small.

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

Is there $f$ for which:

1 Both $x,y$ are squares infinitely often, i.e. $x=u^2,y=v^2$?

2 Both $x,y$ are (arbitrary) large powers infinitely often: $x=u^k,y=v^m$ for some $k,m \ge 2$ (the larger the better)?

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