You can skip submitting a phone number. They don't make it obvious but an option to skip is there at sign up. Though from personal experience it may make the moderation system more aggressive towards your account.
Thanks, I did skip it on signup, but now every time I try to log in, I'm forwarded to a phone entry page, and there's no option to leave it except by signing out or hitting the help page. It appears I'm worthless to Twitter without my phone number, something I'm not willing to give them.