Does pure iron not rust? In my school science textbook, written more than twenty years ago, it said that pure iron does not rust. Accompanying was a photograph of an ancient iron statue situated outside, which had not rusted.

Is this correct?
The purity of the iron is less important here than the conditions of the atmosphere and other surroundings.  Rusting of iron involves the formation of hydrated oxides and so requires the presence of moisture, either as water vapor or as liquid.  In a dry environment iron is indeed more resistant to corrosion because it forms anhydrous oxides which, as long as moisture is kept away, do provide some passivation.