Solution of Waring's Problem Mod n

Sums of Powers in Large Finite Fields

“…For k = 3, 4, 5 the Theorem implies that two kth powers suffice as soon as q > 16, 81, 256, respectively, whereas the largest prime fields requiring three kth powers have 7, 41, 101 elements, respectively. These computations, as well as the above Theorem in the prime-field case, were noted in [6] and [7]. For k < 3 the Theorem is not new.…”

“…Nagell [4] showed that the field with seven elements is the only prime field containing elements which are not sums of two cubes. A different proof, based on a theorem of Vosper, appears in [6]. There is also an older proof due to Skolem [5], based on a result of Hurwitz [1].…”

Waring's number mod m

“…As noted by Small [15,16] the main difficulty in going from representations of a number as a sum of kth powers (mod p) to representations (mod p n ) is in dealing with values of k divisible by a power of p that prohibits the lifting of solutions. Small gives a procedure for determining the value of γ (k, p n ) and calculates the value for a number of special cases including k = 2 and 3.…”

“…Then γ (k, 2 n ) = 2 n − 1 if 4 n e + 2, and γ (k, 2 n ) = 2 e+2 if n e + 3 and k 6. Small [15,16] had already treated the cases k = 2 and 4: γ (2, 2 2 ) = 3, γ (2, 2 n ) = 4 for n 3, γ (4, 2 3 ) = 7, γ (4, 2 n ) = 15 for n 4. Henceforth, we shall assume p is odd.…”

A new variant of the Hilbert-Waring problem

“…There are a variety of generalizations or variations for the Hilbert-Waring problem, e.g., g(k, m) is defined by C. Small [4] as the smallest positive integer s such that every integer is a sum of s k-th powers mod m, i.e., (2) n…”