23

This number is a prime.

Every positive integer is the sum of at most nine cubes. Only 23 and 239 require all nine! Waring's problem (1770) asks: if given k, is there some number g(k) for which every positive integer can be written as the sum g(k) (or fewer) kth powers? For example, g(3) = 9 and g(4) = 19. In 1909, Hilbert proved the answer was yes.