Leonard Eugene Dickson studied generalizations of
Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers[2] (17 is 1, and 27 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers.[3] If powers of negative integers are allowed, only 12 powers are required.[4]
The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800.[5]
The smallest seventh power that can be represented as a sum of eight distinct seventh powers is:[6]
The two known examples of a seventh power expressible as the sum of seven seventh powers are
^Kumchev, Angel V. (2005), "On the Waring-Goldbach problem for seventh powers", Proceedings of the American Mathematical Society, 133 (10): 2927–2937,
doi:10.1090/S0002-9939-05-07908-6,
MR2159771