Last number requiring 37-n positive fifth powers to represent.

223, 222, 221, 220, 219, 466, 465, 464, 463, 952, 2999, 6123, 7138, 7738, 10768, 14914, 22625, 30220, 32016, 87918, 98604, 312389, 470348, 786159, 2103306, 6293040, 51033617, 617597724, 68578904422

1

There are two additional very large terms which are unknown at this time. Jingrun proved that 37 terms is enough to represent any non-negative number.

T. D. Noe, Plot of 29 terms

Chen Jingrun, On the estimation for g(5) in Waring’s problem, Sci. Rec; (1959), 327–330.

Wikipedia, Waring’s problem

nonn,fini

T. D. Noe, Jul 04 2017