You potatoes will not last as long in the refrigerator as they will in cool,  humid, dark location. The starch does change to sugars at refrigerator temperatures.  
To store your potatoes, do not wash and inspect for soft spots, mold, scares and pest damage.  
Best to store them in a container that provides ventilation such as mess bags, cardboard boxes or ever paper bags.  Avoid plastic bags.  
Store in a cool, dark, humid location (45-55 degrees F.).  

It is a good idea to cure your potatoes before you store them and do not wash them.  
  
Don't store them with onions or apples.  They put off a gas that ripens them more quickly.  

Check them often and pull out those that are starting to spoil. 

White and yellow potatoes tend to store longer than the red potatoes.   

Below is some information on refrigerating potatoes from the Idaho Potato commission. 
 

Refrigerator temperatures are too low, which converts the potato’s
starch into sugar, resulting in a sweet taste. The extra sugar also
causes potatoes to darken prematurely while frying. (This process can
sometimes be reversed by storing the potatoes at room temperature for a
week to 10 days.) If potatoes have green patches, cut them off, as they
have been exposed too long to direct lighting and will have a faintly
bitter taste. The rest of the potato will taste fine.

Potatoes are still alive when they are picked, so they should be kept
in a cool area, away from the light. A cellar is ideal, but any place
where they won’t be exposed to excessive heat or light will help prevent
spoiling.

Here is a publcation on storing potatoes from the University of Idaho Extenion Service that you might find helpful.  

https://www.uidaho.edu/-/media/UIdaho-Responsive/Files/Extension/publications/cis/cis1153.pdf?la=en

Hope that helps.  Good luck storing your potatoes.  

Thanks for using Ask Extension

Nellie Oehler