It is given two RSA moduli $N_1$ and $N_2$, known to be of the form $N_i=p_i\cdot q_i$, with $p_i$ and $q_i$ unknown primes, and such that $p_2=p_1+2$. Can we make use of that relation to factor the moduli much more efficiently than if it did not held?
If that helps, add extra assumptions if they are plausible for common RSA moduli, such as:
- $p_1<p_2<q_1<q_2$
- $p_1\equiv1\pmod 4\;$ so that $p_1$ and $p_2$ differ only by their second lowest-order bit
- $2^{(k-1)/2}<p_i<q_i<2^{k/2}$ with $k=1024$
This is a minimal subset of this more general question.