For me, the fastest way to gain pagerank was to put a 301 redirect from my old site with pagerank, to the new one which didn't have any. In my case, I was moving to a different domain, but I suppose you could just buy an old expired domain with pagerank, redirect it to your domain and get the same result.
Page rank of 4 in 3 weeks is done by launching your site 3 weeks before the PR update and getting a solid PR 5 link during those 3 weeks. So launch your site 3 weeks before the PR update and get a link in site-sift. That is how it is done.
One could speculate, if PageRank truly is on a logarithmic scale, that the acceptable frequency of new links grows exponentially, along with site age and PR.
That possibilty heightens my concern. All sites ahead of me in the SERPs have been online longer, and therefore, would be allowed to gain more links per day or month, or whatever. The sites ahead of mine in the SERPs already have a huge advantage, in that, they are attractive to link to. If there's a limit to gaining links imposed on new sites, we have no chance of ever catching the guy that already has 800 backlinks and is gaining more everyday by invitations. Bompa
I suppose that is true in theory, as with any other form of race where one contestant has a head start... However, that requires the 'other guy' to perform at his best, not allowing for anyone to catch up. Simply put, you just have to do better than him, gaining better links with more value and paying attention to all the other important factors. That's the real challenge of SEO - doing a better job than the next guy