RSA - Very big Modulus and Ciphertext
2466 digits for an RSA modulus is somewhat large, but not unseen. A larger RSA modulus, when it is properly sampled, can only lead to a better security guarantee. The ciphertext in RSA is $c = M^e \bmod n$, where $M$ is something derived from the input message $m$ - hence, it is perfectly normakl for $c$ to have the same bit length as $n$ (anything significantly smaller would be a weakness, anything significantly larger... would not be a valid ciphertext). This choice of $e$ is also the most common one. Hence, there is nothing wrong with your proposed choice of parameters, provided that $n$ was indeed generated as the product of two random length-$1233$ strong primes.


  n cannot be factorized. I checked on factordb.com and no factors were present.


Note, however, that this gives you essentially zero indication about the actual security of this modulus. It is only a secure modulus if it was generated as a product of two equal-length random strong primes, with a good random number generator. $n$ could be a very weak RSA modulus if it was not generated this way, yet you would not immediately find the factors with an online tool.