The BOINC daemon uses port 80 (http) for communications between the host and project server. With 5.1.x http will be joined by https.

The BOINC Manager uses ports 1043 and 31416 for communications with the BOINC daemon. 1043 is the primary port (registered) and 31416 is the fallback (not registered).

BOINC.exe (the daemon) must be allowed to act as a server on ports 1043 and 31416, and also access the internet on port 80 (and whatever https is). The usual problem is that there is a personal firewall that is being overzealous.

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BOINC WIKI