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.

Nano, formerly known as Railblocks, is a cryptocurrency that is focused on enabling fast transactions with no fees. Nano's innovative blockchain architecture utilizes a block lattice structure that does not need miners to confirm transactions.
Nano uses a Delegated Proof-of-Stake consensus.

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Sharechain is developing a Proof-of-Work cryptocurrency platform. Their network is currently fueled by the Ethereum-based SSS token. Their marketing materials are unclear and do not offer any information about its advantages.