Powersums representing residues mod p^k, from Fermat to Waring
Nico F. Benschop · arXiv (Cornell University) · 2001
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 n^p residues 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 p^2, 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 2}) of divisors of p^2-1 are derived. -- [Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261]