Every integer greater than 454 is the sum of at most seven positive cubes

Abstract

A long-standing conjecture states that every positive integer other than

$$\begin{array}{cc}15,22,23,50,114,167,175,186,212,& \\ 231,238,239,303,364,420,428,454& \end{array}$$

is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on the basis of extensive calculations performed by the famous computationalist Zacharias Dase. We complete the proof of this conjecture, building on previous work of Linnik, Watson, McCurley, Ramaré, Boklan, Elkies, and many others.