lol is there a link where you can give proper documentation here?
This is not true.. some of the sites on the top 1 position have Pr 0-2 when competitiors are PR5++
But just to answer your question on how to increase page rank get High Page rank Backlinks everysingle day and force index them.. that's based on experience. no one can exactly tell you how to increase page rank or give you a proper documentation for it cause only google knows exactly how..
You are right & wrong. PageRank (note the absence of a " " <space>) does not determine rank on the SERP (Search Engine Results Page), however Google does "use" (quoting the OP) "PageRank to rank web pages", it's just not the ONLY thing. It's one of 200 factors (old information, may no longer be true), and it's been said that PageRank has been deprecated and is no longer as important a factor as it used to be (new information, and may also not be true).
Which is why you see PR2 site ranking higher than PR5 sites.
But all of this is meaningless, because it ignores the most important, end-all, be-all quality and that is,
what does Google think is most relevant to the particular keyword? RELEVANCE is the end-all, be-all. It's the whole point of the algorithm. Everyone gets all spun up over this factor and that factor, and all these squabbles blow up and no one pays attention to the most important fact which is at the end of the day, Google and it's algorithm delivers what it thinks is relevance to any given keyword.
Which means that it is a waste of time to focus on any particular variable without also taking into account, not only all the other 199 variables, but also how all those variables compare with all the other websites that have that particular keyword.
So you cannot possibly have a simple conversation between people that "sort of" know this stuff, because there are too many variables in play to allow for any kind of basic discussion. It should all be couched in open-ended terms like "might" and "could" and also anecdotal examples such as "well, I tried this and then this happened, so maybe...." The simpler the assertion is, and the more confident the person saying it sounds, the more likely it is to be completely wrong.