Finding the first few digits of p and q
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.