If it were me I would buy a Trail Watcher camera (www.trailwatcher.com) with rechargeable AA's.

The rechargeable AA's will take over 400 pictures and not having any problem lasting for a month. The number of pictures will wear them down before the time will. You just need to make sure you set the picture delay long enough so that it will last four weeks. You will have to experiment with the time to see what you will need. It will be a pretty long delay for four weeks, i.e. several minutes.

So far I've tried all of the major brand names, not every model, and yes there are several cameras out there that could take more pictures with a bigger card and solar power but I've not used one that compares in quality and dependaility to the Trail Watcher.