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I'm trying to run a remote kernel, but my local system administrator is wary of opening up ports for incoming connections.

Is there are particular range of ports that Mathematica uses to communicate with remove kernels?

Even if it's a wide range (e.g. 40000-70000), such an answer would still be helpful to me.

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Most likely it's the entire ephemeral range. This is usually controllable by the OS, although you should be aware that reducing it to a much narrower range may cause problems if you need to open a lot of network connections simultaneously. –  Oleksandr R. Aug 8 '13 at 7:56

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