It's not about background size. It's about how you design your game to fit mobiles screens.
If your game is 640x320, and it's scaling to fit mobile entire screen, then there's no point of using full HD image for the background.
Or maybe you are asking for the background size as a layout size, of the game? You need to clearly describe what you want to get best answers.
So, the game will work only in landscape mode.
i need hmm a size that can fit on smartphones(i will use scaling to work well on iphone4s and a galaxy S5 for exemple)