User blog comment:PsiCubed2/CURRENT VALUE OF PEGG/@comment-27516045-20170518205217/@comment-10429372-20170526151103

This isn't too difficult to prove. 81 and larger fourth powers won't work, as they are too large. So we are left with 16s and 1s only, and we can have only 4*16. Then we need 15*1, so 19 in total. Decreasing the number of 16s will only increase the number of fourth powers.