Determining interval(s) containing the smallest prime $p$ of an RSA modulus? Are there any results that have determined interval(s) $$b\sqrt{n}&lt;p\leq a\sqrt{n},$$ where $0&lt;b&lt;1$, $0&lt;a\le 1$, and $p&lt;q$ are primes such that $n=pq$,

thus allowing us to narrow the search for $p$?
Any such nontrivial intervals would be entirely up to the implementation generating the RSA key.  For example, when generating a 2048 bit RSA key, an implementation could generate a 512 bit $p$ and a 1536 bit $q$; such a modulus would work just as well in RSA as any other modulus; this gives us an $a \approx 2^{512}$.

However, while an implementation could do that, they never do; instead, they generally aim to keep $p$ and $q$ balanced; one advantage this has is that it allows the maximum speed-up by the CRT algorithm.

I've seen two different strategies in common use; if we assume $n$ is even, then to generate an $n$-bit RSA modulus, we often either:


Generate two random $n/2$ bit primes with the two high bits set (which ensures that, when $p$ and $q$ are multiplied together, yield a $n$-bit modulus, and not an $n-1$ bit.
Generate two random primes both from the range $(2^{(n-1)/2}, 2^{n/2})$, which yields the same effect, albeit in a slightly more mathematical way.


With the first strategy, we have $\sqrt{3/4}\sqrt{n} &lt; p &lt; \sqrt{n}$; in the second strategy, we have $\sqrt[4]{1/2}\sqrt{n} &lt; p &lt; \sqrt{n}$ (a slightly wider interval).

As I mentioned earlier, there's no requirement that any specific RSA implementation uses either strategy (unless they claim FIPS compliance; FIPS insists on the second strategy).

In addition, you appear to be interested in this to help in searching for factors of RSA modulii; for any RSA modulii that's even slightly secure, the interval is far too wide to make any such simple-minded search for factors feasible.