I don't know if peeling and/or chopping up the potatoes will help prevent them from sprouting but it will result in the potatoes becoming inedible much faster. The skin helps keep the moisture in the potato and protects it from pathogens. If you peel them the exposed surface will dry out quickly and they're no longer safe to store at room temperature. They should be safe for a few days stored in fridge, but that won't prevent them from drying out.

Fortunately as the answers to the question [Is it safe to eat potatoes which have sprouted?][1] you don't have to worry about your potatoes sprouting. As they don't go soft they're still safe to eat.

If you're finding that your potatoes are actually going bad before you can eat them you could freeze them, but this probably isn't worth the trouble. Just buy fewer potatoes.

  [1]: http://cooking.stackexchange.com/questions/1260/is-it-safe-to-eat-potatoes-which-have-sprouted