Can RSA encryption produce collisions? In RSA, a message is encrypted by $m^e \pmod N$. $N$ is the modulus, $m$ is the message and $e$ is the public exponent. (I know that $m$ should not be greater than $N$.)

My question is, can $m^e$ be greater than $N$ (obviously, before taking the modulus)?

In that case is there a possibility like  $ m_1^e=m_2^e \pmod N$, i.e. can we get a collision?
Correction to Henricks answer: collisions are impossible (unless someone did something wrong).  That is, if:


$e$ is a proper RSA exponent (that is, relatively prime to $p-1$ and $q-1$, where $p$ and $q$ are the factors of $N$), and:
$m_1 \neq m_2 \mod N$ (that is, you're not trying to encrypt the same message twice),


Then we will always have $m_1^e \neq m_2^e \mod N$

This is rather implied by the fact that the RSA operation can be inverted using the decryption exponent; if two different messages collided, then that couldn't be inverted uniquely.