  The availability of working quantum computers has led to several proposals
and claims of quantum advantage. In 2023, this has included claims that quantum
computers can successfully factor large integers, by optimizing the search for
nearby integers whose prime factors are all small.
  This paper demonstrates that the hope of factoring numbers of commercial
significance using these methods is unfounded. Mathematically, this is because
the density of smooth numbers (numbers all of whose prime factors are small)
decays exponentially as n grows. Our experimental reproductions and analysis
show that lattice-based factoring does not scale successfully to larger
numbers, that the proposed quantum enhancements do not alter this conclusion,
and that other simpler classical optimization heuristics perform much better
for lattice-based factoring.
  However, many topics in this area have interesting applications and
mathematical challenges, independently of factoring itself. We consider
particular cases of the CVP, and opportunities for applying quantum techniques
to other parts of the factorization pipeline, including the solution of linear
equations modulo 2. Though the goal of factoring 1000-bit numbers is still
out-of-reach, the combinatoric landscape is promising, and warrants further
research with more circumspect objectives.
