  This paper builds on top of a paper we have published very recently, in which
we have proposed a novel approach to prime factorization (PF) by quantum
annealing, where 8,219,999=32,749x251 was the highest prime product we were
able to factorize -- which, to the best of our knowledge is the largest number
which was ever factorized by means of a quantum device. The series of annealing
experiments which led us to these results, however, did not follow a
straight-line path; rather, they involved a convoluted trial-and-error process,
full of failed or partially-failed attempts and backtracks, which only in the
end drove us to find the successful annealing strategies. In this paper, we
delve into the reasoning behind our experimental decisions and provide an
account of some of the attempts we have taken before conceiving the final
strategies that allowed us to achieve the results. This involves also a bunch
of ideas, techniques, and strategies we investigated which, although turned out
to be inferior wrt. those we adopted in the end, may instead provide insights
to a more-specialized audience of D-Wave users and practitioners. In
particular, we show the following insights: ($i$) different initialization
techniques affect performances, among which flux biases are effective when
targeting locally-structured embeddings; ($ii$) chain strengths have a lower
impact in locally-structured embeddings compared to problem relying on global
embeddings; ($iii$) there is a trade-off between broken chain and excited CFAs,
suggesting an incremental annealing offset remedy approach based on the modules
instead of single qubits. Thus, by sharing the details of our experiences, we
aim to provide insights into the evolving landscape of quantum annealing, and
help people access and effectively use D-Wave quantum annealers.
