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Primecoin is a cryptocurrency based on finding prime numbers. Identifying prime numbers plays a central role in cryptography. Primecoin's unique Proof-of-Work (PoW) consensus mechanism is able to increase transaction speeds and improve energy efficiency.
The network searches for special prime number chains known as Cunningham chains and bi-twin chains, which are of interest in mathematical research.

FREE Coin is an Ethereum-based token with a focus on being inclusive and encouraging cryptocurrency adoption. Their goal is for the token's price to increase in value, until its price is close to the USD. FREE Coin's site does not provide any details about their development plan or what specific advantages their network will provide.