Waring's problem for sums of fourth powers of positive integers: \(g(1,4)=21\) (Q1896936)

The author's main result is a recurrence estimate for the function \(G (m,r)\) by the function \(g(m - 1,r)\), both of which occur in the generalized Waring problem (with the summands to be \(\geq m)\). A consequence of this is that \(g(1,4) = 21\), that is, that any natural number, except for a finite number which can be computed explicitly, can be represented by a sum of 21 fourth powers of positive integers. The author and Vladimir Voevodskij spent a month of computer time to find that the number 77900162 can be simultaneously represented as a sum of \(2,3,4, \dots, 20\) and 21 fourth powers of positive integers. There are also analogous results for sums of squares and cubes.