How can I take advantage of repeated patterns in non random RSA prime factors?
Theorem 1 of "Implicit Factoring with Shared Most Significant and Middle_Bits" by Jean-Charles Faugère, Raphaël Marinier, Guénaël Renault from 2010 deals with factoring products of two primes where some fraction of the MSBs are common to one of the primes in each of the composites. If I read their bounds correctly, their method must only be useful for quite unbalanced primes.

"Let $N_1=p_1*q_1,N_2=p_2*q_2$ be two n-bit RSA moduli, where the $q_i$s are $\alpha$-bit primes and the $p_i$s are primes that share the $t$-most significant bits, If $t&gt;2*\alpha+3$ then $N_1$ and $N_2$ can be factored in time quadratic in $n$"