Siemens 4100 modem

My modem will disconnect instantly with the briefest of power interruptions. When power is restored, the internet is lost. I have had to reinstall the Linksys router to restore internet in the past. I have a new Linksys router, BEFSX41, and the problem continues. ATT&T tech support checked the outside drop and they say all is well outside. I have checked that all of the DSL filters are in place and replaced phone lines inside. Still, I must restart/reset the modem, router, and computer multiple times before internet access is restored. Do I have a bad modem?
Thanks