I am accross this equation with no known analytical solutions: x^5 - x + 1 = 0. I asked this before and the answer was that you can get as good an approximation you want by approximation methods. It is possible that when applying approximation methods, you will get singularities. These methods won't work. The equation is bullet proof. The question is can we be cetain that there are approximation methods that will work or is this equation intrincally unvolvable even by approximatiuon methods? My second question is messy. It is a stray thought. Is this equaton one that satisified Godel's criteria for an unsolveable equation is tue but can be shown to be not provable and not unproveable? If it fits Godel's criteria it would be nice to show studends. PS: Anyway, In teaching algebra, it is nice to show this unsolveable simple equation.