  We introduce a new deterministic factoring algorithm, which could be
described in the cryptographically fashionable term of "factoring with hints":
we show that, given the knowledge of the factorisations of
$O(N^{1/3+\epsilon})$ terms surrounding $N=pq$ product of two large primes, we
can recover deterministically $p$ and $q$ in $O(N^{1/3+\epsilon})$ bit
operations. Although this is slower than the current best factoring algorithms,
this method shows that the factorisations of close integers are related and
that consequently one can expect more results along this line of thought.
