Tight Bounds on the Sum and Difference of Divisors of RSA Challenge Numbers The title says it all: given that $n$ of a given length $d$ is a RSA Challenge Number where $n=pq$, where $p$ and $q$ are two primes of length $d/2$. My question is, knowing how the RSA numbers are generated, what are some tight upper and lower bounds of the sum and difference of $p$ and $q$? I'm asking this question as I am working on a factoring algorithm and it works best with tight bounds.
EDIT: For an example, I am specifically interested in RSA-2048.

Well, it is known that if the primes $p,q$ have approximately the same length, then there is a simple attack called Fermat factoring.
To this end, note that one can write $u=(p+q)/2$, $v=(p-q)/2$, $p=u+v$, and $q=u-v$ if $p>q$.
If $p,q$ are close together, then $v=(p-q)/2$ is small and $u$ is only slightly larger than $\sqrt n$.
Then use the algorithm to start from $u=\lfloor\sqrt n\rfloor +1$ and add to $u$ until you find that $u^2-n$ is a perfect square. Then $v=\sqrt{u^2-n}$ and $p,q$ are given as above. Done.
