How much does it cost to buy a good Google PageRank?

The PageRank algorithm is used by the popular Google search engine to rate website reputation. It is one of the key ingredients that affects placement in search results coveted by webmasters. I prove that attempts to manipulate a site’s PageRank by creating “link farms” are futile, and that PageRanks can only be increased by either recommendations from reputable websites or by buying domain names. I derive a tight lower bound on the cost of the latter technique. I use these theoretical results to estimate what this lower bound means in $US at the time of writing. I also discovered a mistake in the definition of PageRank in the original paper. This is discussed in the appendix.