Primecoin has come far in the last few days. Less than a day after founder Sunny King released the genesis block and I first covered the currency, Primecoin has become one of the few alternative cryptocurrencies to be featured on Business Insider, its thread on Bitcointalk has reached 77 pages and has attracted the interest of thousands of people around the world. Now, after two days in the wild, Primecoin is already bringing us real-world value.

Buying primecoins has already become easy; Coins-e, a cryptocoin exchange featuring over 30 currencies, added Primecoin support yesterday. Users need only create an account, deposit bitcoins, and within one hour they will be able to exchange them for primecoins at a current rate of 1 for 0.005 BTC. And if you want to spend them, there’s a business that accepts those too: Satoshi Roulette. SatoshiRoulette is a Bitcoin gambling site that popped up a few months after the industry was first popularized by Erik Voorhees’ SatoshiDice, featuring a roulette game that you can play simply by going to the site, clicking on the guess you want to make (either a number, like 15 or 23, or special options like “FIRST 12″, “EVEN” or “3rd line”) and sending money to the Bitcoin address. As far as infrastructure is concerned, there is now a Primecoin block explorer, which can theoretically be used by web developers to easily set up a Primecoin-accepting store.

However, that is not even the most interesting aspect of Primecoin’s development. Rather, what deserves the most mention of all is that Primecoin miners have already broken world records for finding Cunningham chains. The specific record is for Cunningham chains of length 9; the longest three chains before this week had origins 111, 109 and 94 digits respectively, and had been discovered in 2004 and 2009. The two new records generated this week are 131 and 158 digits respectively. The numbers:

15746436948707162347681191092622506406480521554672382866529

11336545168633619161630382098525129584202068464052357630633

6183730859560











65004063504559525007738276505391185322137155270201199057971

07651195404566504896517372220950102661115676898877913886840

8914387452650147975793533060274322762630







Note that these numbers are obviously not prime; the convention of Cunningham and bi-twin chains (or at least the convention adopted by Sunny King) is that the “origin” of a chain is defined as either one less or one more than the first prime in the chain. This is done for convenience. If one wanted to retrieve some particular prime in the chain from the first prime, one would need to repeatedly multiply by two and subtract one at each step; using this convention, the formula is simply p n = w*2n-1, where n is the index of the prime and w is the origin. For Cunningham chains of the first kind, the formula would be p n = w*2n+1. Thus, the first prime in the smaller record-breaking chain is:

15746436948707162347681191092622506406480521554672382866529

11336545168633619161630382098525129584202068464052357630633

6183730859561







The second prime is:

31492873897414324695362382185245012812961043109344765733058

22673090337267238323260764197050259168404136928104715261267

2367461719121







And so on.

A list of records is being maintained by Dirk Augustin; the miners of blocks 2044 and 5355 have the option of proving who they are (eg. by sending a pre-specified quantity of primecoins from that address) in order to get their names on the list. Length-9 Cunningham chains of the first kind will be much harder to break records for; the current record is already 185 digits long. Cunningham chains of length 10, however, will be easy; the current records are 99 and 109 digits long, respectively. Ultimately however, the Primecoin communities should set their sights on a much more audacious goal: the first advance in mathematics made for the sake of Primecoin mining. In a few years, or perhaps even sooner, we may well get there.