From Fermat to Waring

The ring Zk(+, .) mod p k with prime power modulus (prime p > 2) is analysed. Its cyclic group Gk of units has order (p − 1)p, and all p-th power n 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 p, consisting of p − 1 residues with n ≡ n mod p. The concept of carry, e.g. n in FST extension n ≡ np + 1 mod p, is crucial in expanding residue arithmetic to integers, and to allow analysis of divisors of 0 mod p. For large enough k ≥ Kp (critical precison Kp < p depends on p), all nonzero pairsums of core residues are shown to be distinct, upto commutation. The known FLT case1 is related to this, and the set Fk + Fk mod p k of p-th power pairsums is shown to cover half of Gk. Yielding main result: each residue mod p is the sum of at most four p-th power residues. Moreover, some results on the generative power (mod p) of divisors of p± 1 are derived. [Publ.: Computers and Mathematics with Applications V39 N7-8 (Apr.2000) p253-261] MSC classes: 11P05, 11D41