Primecoin

The innovative prime proof-of-work in Primecoin not only provides security and minting to the network, but also generates a special form of prime number chains of interest to mathematical research. Thus primecoin network is energy-multiuse, compared to bitcoin network. Primecoin is designed to sustain a prosperous mining market and high level of security, while maintaining good scarcity property like gold. Primecoin also processes payment transactions 10x faster than bitcoin network.

Primecoin network searches for special prime number chains known as Cunningham chains and bi-twin chains. The distribution of these prime chains are not well-understood currently as even for its simplest case twin primes their infinite existence is not proven. The distribution of primes has been one of the most important discoveries in arithmetic, and the study of prime chains traces its lineage to the work of Riemann and prime number theorem, with connections to the deeper nature of the seemingly random pattern of prime distribution. Prime distribution is not just an abstract interest of mathematicians. Riemann's study revealed connections between Riemann zeta function and prime distribution, whereas later on Riemann zeta function has been shown to be highly relevant in other scientific disciplines such as physics, thus the study of prime distribution is an important part of the foundation of modern sciences.

References on the purpose of searching for primes and prime patterns in general:

• Structure and randomness in the prime numbers - presentation by Terence Tao
• Why do people find primes
• Ask Ars: Why spend time and money finding new prime numbers?