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This paper studies the optimal clearing problem for prosumers in peer-to-peer (P2P) energy markets. It is proved that if no trade weights are enforced and the communication structure between successfully traded peers is connected, then the optimal clearing price and total traded powers in P2P market are the same with that in the pool-based market. However, if such communication structure is unconnected, then the P2P market is clustered into smaller P2P markets. If the trade weights are imposed, then the derived P2P market solutions can be significantly changed. Next, a novel decentralized optimization approach is proposed to derive a trading mechanism for P2P markets, based on the alternating direction method of multipliers (ADMM) which naturally fits into the bidirectional trading in P2P energy systems and converges reasonably fast. Analytical formulas of variable updates reveal insightful relations for each pair of prosumers on their individually traded prices and powers with their total traded powers. Further, based on those formulas, decentralized learning schemes for tuning parameters of prosumers cost functions are proposed to attain successful trading with total traded power amount as desired. Case studies on a synthetic system and the IEEE European Low Voltage Test Feeder are then carried out to verify the proposed approaches.