Does anyone know a good way to store potatoes? I love the things and would like to keep a large quantity on hand, but after a few weeks they begin to rot.

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it is important to pick the potates mature first,those are the potatoes that can stored for months,then dont rinse before storing, pack it in brown papers with holes,not preferable to use plastic ones.make sure to keep the stored potatos in a cool (not cold),dry place. it can be kept for months,take care to pick and remove soft ones from time to time and sprouted ones too.
Have fun

Potato is a delicate thing that must be stored with care. The first thing is not to rinse them and to store them at a optimum temperature of 30 to 40 degree Celsius. And don't keep potato freeze for more than 50 days.Remove sprouts in them by checking them once a week.

we have a storm/root cellar that's 10 feet underground.

I make sure the potatoes are good and mature, with no blemishes at all, then I spread them out on newspaper in the dark cellar for a few days (don't wash or brush them). I gently pack them in orange crates, inspecting them for soft spots (which will show up easily).

My potatoes usually last till the next summer's new potatoes are coming on. It's IMPORTANT to let them dry out dirty. Damp potatoes will grow mold.

BTW, Sweet potatoes must dry MUCH longer.

Just a note here: our root cellar has an exhaust fan and an air inlet duct down by the bottom of the door. You must dry potatoes in a dark well ventillated place. If it's light, the skins will turn green, then they will spoil.