Powersums representing residues mod pk, from Fermat to Waring

AbstractThe ring Zk(+,.) mod pk with prime power modulus (prime p > 2) is analysed. Its cyclic group Gk of units has order (p − 1)pk−1, and all pth power np residues form a subgroup Fk with ¦Fk¦ = ¦Gk¦/p. The subgroup of order p − 1, the core Ak of Gk, extends Fermat's Small Theorem (FST) to mod pk>1, consisting of p − 1 residues with np = n mod pk. The concept of carry, e.g., n′ in FST extension np−1 = n′p + 1 mod p2, is crucial in expanding residue arithmetic to integers, and to allow analysis of divisors of 0 mod pk.For large enough k ≥ Kp (critical precision Kp < p depends on p), all nonzero pairsums of core residues are shown to be distinct, up to commutation. The known FLT case1 is related to this, and the set Fk + Fk mod pk of pth power pairsums is shown to cover half of Gk. Yielding main result: each residue mod pk is the sum of at most four pth power residues. Moreover, some results on the generative power mod pk>2) of divisors of p ± 1 are derived