What is the minimum number of socket port(s) required for a TCP server?

Typically one port/socket will be open on the server to listen for incoming connections. Once connected, the connection will have its own (different) socket allocated on the server to continue the conversation. So strictly speaking that's one port and two sockets on the server -- and one on the client if you wish to count that.

Some protocols (like ftp) uses more than one port (one for control and one for data), but TCP in general can use one port at each sides.

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