# What would be the safety requirements for the primes in $n=p \cdot q$ regarding the factorization?

Let it be $$p, q \in \mathbb{P}$$ with $$p,q \in [2^{b-1}, 2^b]$$ for some $$b \in \mathbb{N}$$ and $$p \cdot q = n \in \mathbb{N}$$. What would be the distance between $$p$$ and $$q$$ (as a function of b) so that the factorization of $$n$$ is hardest or be considered difficult?

• Related question. I still think the present question is interesting when it asks for a minimum $\lvert p-q\rvert$ so that the factorization of $n$ can be considered difficult. I don't know if/how the FIPS 186-4 bound $\lvert p–q\rvert>2^{b–100}$ is justifiable (note: it's stated for $p,q\in[2^{b-1/2},2^b]$ )
– fgrieu
Aug 24, 2021 at 7:58