Abstract : The ring Z_k(+,.) mod p^k with prime power modulus (prime p>2) is analysed. Its cyclic group G_k of units has order (p-1)p^{k-1}, and all p-th power residues n^p form a subgroup F_k with |F_k| = |G_k| / p. The subgroup of order p-1, the core A_k of G_k, extends Fermat's Small Theorem (FST) to mod p^k>1, consisting of p-1 residues with n^p = n mod p^k. The concept of carry, e.g. n' in FST extension n^p-1 = n'p +1 mod p², is crucial in expanding residue arithmetic to integers, and to allow analysis of divisors of 0 mod p^k.

For large enough k \geq K_p (critical precison K_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 F_k + F_k mod p^k of p-th power pairsums is shown to cover half of G_k. Yielding main result :
. . . Each residue mod p^k is the sum of at most four p-th power residues.

-- V1: 25oct99, V2: 13may03.

Full text (.pdf). . . . . . 9 pgs (149k) - (c) May'03
Intro (.htm) . . . Semigroups and Arithmetic (re: Fermat - Waring - Goldbach)
FST mod p^3 (.pdf). . . 9 pgs (165k) - (c) Feb'04
. . . "Extend Fermats Small Theorem to r^{p-1} mod p^3 for the divisors r of (p +/- 1)"

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