Astronaut Mark Watney survived on potatoes when he was stuck on Mars in the movie The Martian, but that was fiction. The Great Famine in Ireland came about because of potato blight, and although lack of potatoes caused starvation, they weren't the sole food eaten by the Irish. Andrew Taylor ate a potato-only diet in 2016, but he flavored them with other ingredients and took supplements. Could someone live off only potatoes of they had to?
Technically, the traditional white potato contains all the essential amino acids you need to build proteins, repair cells, and fight diseases. And eating just five of them a day would get you there. However, if you sustained on white potatoes alone, you would eventually run into vitamin and mineral deficiencies. That's where sweet potatoes come in. Including these orangey ones in the mix—technically, they belong to a different taxonomic family than white potatoes—increases the likelihood that the potato consumer will get their recommended daily dose of Vitamin A, the organic compound in carrots that your mom told you could make you see in the dark, and Vitamin E. No one on a diet of sweet potatoes and white potatoes would get scurvy, a famously horrible disease that happens due to a lack of Vitamin C and causes the victim’s teeth to fall out.
But there are drawbacks to potatoes as a life-sustaining food. Personally, anytime someone asks the question, "If you could eat only one food, what would it be?" the answer is supreme pizza. However, that is technically more than one food, which is what you need. Read about the nutritional bang for your buck in potatoes at Popular Science.