Finding Private key in RSA with public key, cipher text and plain text Is there a known 'non-brute force' method of determining a private key in an RSA system when all other parameters are know?

I found the values of a ciphertext ($C$), its corresponding plaintext ($P$) and the values of the public key ($e$,$N$) are known.

We know that: 
$$P = C^d \mod N$$

Here, the value of $P$ is known (the cipher text has been decrypted) and the values of $C$ and $N$. Given these parameters, is it possible to find $d$?

Also, how do I find an integer $k$ such that $(k\cdot N - P) = C^d$ ?

Any suggestions/discussions are welcome.
No, there is no known efficient method to find an RSA private key from public key, ciphertext and plaintext, including when no padding is used and $e=3$. The best known method is factoring the public modulus, which then trivially allows to find a working private key. Factoring is considered brute force, or tantamount to that. The best known factoring method for a public modulus of cryptographic interest in RSA is GNFS.

We can prove the assertion in the first two sentences, modified by adding that the ciphertext was not chosen by an adversary, and that the hypothetical method works including for small public exponents $e$. The idea is that any such method could be turned into a factorization algorithm for any $N$ suitable as RSA key: we chose a small odd $e$, which with fair odds is coprime with $\varphi(N)$; we choose some plaintext, and compute ciphertext by whatever method is normaly used in the RSA variant considered; we can now apply our hypothetical efficient method that finds a private exponent $d$; we compute $f=d\cdot e$, which is such that $f\bmod(p-1)=1$ for any prime $p$ dividing $N$, and that's enough to efficiently factor $N$ (see this answer).