Finding the first few digits of p and q Is there a way to find out the first few digits of the factors of the RSA numbers (RSA-1024 or RSA-2048)? 

I do not want to get all the digits but only first 4-5 digits. My question is thus more precisely:

Is there a known, efficient, classical algorithm, that given a composite number, outputs the 4-5 most significant decimal digits of one of the prime factors of this number?
I do not think this is a practical possibility. In order to get the 4-5 most significant bits, the bits of lower significance must be known. This is due to the manner in which numbers are multiplied. The most significant bits of the product do not depend solely on the most significant bits of the factors. Bits of lower significance may have an effect too due to the propagation of carry bits up the product as partial products are summed. Even with some look-ahead optimizations the bits of lower significance would still need to be known. The best I could see would be a probabilistic scheme in which k>5 bits would would be attempted to be solved and the result would be probabilistically correct with some chance as a function of k.