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Twitter says that it will unlock

if I tell it my phone number. So I will no longer be tweeting from there.

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and

there's an alternative way: you file a support ticket and then, I already forgot, either reply to their mail or click their link

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and

the idea seems to be, if they get enough data from you in different protocols, they can be reasonably sure you're not a bot

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and

you may also be able to use a google voice number