[math-fun] Waring's problem for 5th powers

I've seen various older references (Dickson 1931, Herstein & Kaplansky 1978), citing evidence that every positive integer is the sum of no more than 37 positive 5th powers (with, e.g., 223 requiring that many). But I haven't seen a claim that this has been proved. Can anyone please say whether this has been proved or not?* Thanks, Dan _________________________________________________________________________ * I'm also curious about the number proposed h(n) by Herstein & Kaplansky as the Waring number for all nth powers -- has this been proved to be the exact maximum number of nth powers needed to represent any positive integer? But for the moment I'm much more interested in 5th powers.