All the blocks in the Bitcoin block chain have a short string of meaningless data—called a nonce—attached to them. The bitcoin ppcoin qt computers are required to search for the right meaningless string such that the block as a whole satisfies a certain arbitrary condition.
Specifically, it is required that the SHA-256 hash of the block have a certain number of leading zeros. The procedure, remember, is totally arbitrary. It is simply an added complication, like a ritual, so as to make blocks more difficult to generate. Really anything else would do, as long as it was computationally difficult. Other crypto-currencies use other hash algorithms. There is no special condition from number theory which only someone like Shinichi Mochizuki could understand.
Although the purpose of the mining computers is to do the accounting for the block chain, most of the work they actually do is to search for good nonces, rather than anything to do with accounting. The energy used to find the nonces is lost forever. The energy does not “back” the value of bitcoins in the way that gold backs an honest bank note, as some have supposed. Of the vast computing power that goes into bitcoin mining, all but a tiny fraction is apparently purposeless. When a person upgrades their mining computer, they mine at a faster rate and therefore earn more bitcoins.
However, when everyone upgrades, the mining does not become more efficient as a whole. There is only supposed to be one new block every ten minutes regardless of how hard the network is working. Instead, the network updates the difficulty to require more stringent conditions for future blocks. Why tie each bitcoin block to a difficult Procrustean bed?