What problem does this service solve?

Primecoin uses prime number generation to improve Bitcoin's PoW protocol.Apollo aims to integrate a variety of decentralized services in one comprehensive blockchain-based platform.

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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.

Apollo is developing an all-in-one cryptocurrency platform, based on the APL coin. Apollo was created as a fork of NXT, and has a very ambitious plan to integrate a variety of services under one platform, with a strong focus on privacy. Apollo's blockchain is called Hermes and has a Proof-of-Stake consensus mechanism. Some of the features that Apollo's marketing materials claim they are working on are: private transactions with IP masking, smart contracts, encrypted messaging, file sharing, a decentralized marketplace, voting system for governance, decentralized data storage, and a wallet with a built in exchange. John McAfee, a notable blockchain enthusiast and antivirus pioneer, joined the project in October 2018.

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