Technically, its possible - but has nothing to do with permanent residency. You're thinking of naturalisation
Permanent Residency is just where you are of a different "visa" class that has no restrictions on work, requires you to have lived here a number of years already, and is generally given much longer between renewals, if it needs renewal - I forget about the last point...
(And in that case, when you become "Japanese", you get a Japanese passport etc, and you are supposed to hand your American passport in to the US Embassy. Fail to do so and continue to renew it when it expires, and noone really cares, and you continue to get the befits. Just don't flaunt it and its fine.)