I recently created a new twitter account (Damn Nature, You Scary). I had trouble registering a new app as, apparently, each app needs a (unique) mobile number.
The process wouldn’t let me change the permissions of the new app to read and write.
I managed to get this working by:
Download the app on my mobile (iOS in my case, although Android apparently works, too)
Log into the app
Still on your mobile, go to http://dev.twitter.com/apps
Change the permissions of your app to Read & Write
That should work!