Determine RSA modulus from encryption oracle
We have $E(x) = x^{65537} - k \cdot n$, for some integer $k$ (which will be different for different values of $x$), and the unknown modulus $n$, and hence $x^{65537} - E(x)$ will always be a multiple of $n$.

So, compute:

$$\gcd( 2^{65537} - E(2), 3^{65537} - E(3) )$$

That will be $n$ multiplied by some integer which is likely to be small...



As similar method (that doesn't involve computing on such large integers, and even works even if you don't know $e$) is to compute:

$$\gcd( E(2)^2 - E(4), E(3)^2 - E(9) )$$

How this works should be fairly obvious...