RSA generation of private key using public key In RSA private key generation 

e*d ≡ 1 mod φ


e is public, also n is public. How to prove mathematically, generation of private key d is not possible using the same equation and public key e
It isn't impossible. Otherwise, we wouldn't have to keep increasing key sizes of our RSA keys, see this for the history.

As stated in a comment, it is believed to be computationally hard. Though, even that has never been proven.