幂和表示模p^k的剩余，从费马到华林
Powersums representing residues mod p^k, from Fermat to Waring
摘要：
环Z_k（+，.）mod p^k具有素数幂模（素数p>2）。它的单位循环群G_k的阶为（p-1）p^{k-1}，所有p次幂n^p的剩余构成子群F_k，其中|F_k| = |G_k| /p。p-1阶子群是G_k的核A_k，它将费马小定理（FST）推广到mod p^{k>1}，由p-1个剩余组成，其中n^p = n mod p^k。“进位”的概念，例如FST扩展n^{p-1} = n 'p +1 mod p^2中的n'，在将余数算术扩展到整数时是至关重要的，并且允许分析0 mod p^k的除数。. . . .对于足够大的k \geq K_p（临界精度K_p < p取决于p），所有非零的核心剩余对和都是不同的，直到交换.已知的FLT情况_1与此相关，并且p次幂对和的集合F_k + F_k mod p^k被示出为覆盖单位群G_k的一半。--产生主要结果：每个模p^k的剩余是至多四个p次幂剩余的和。此外，还得到了p^2-1的约数的生成幂（mod p^{k>2}）的一些结果. -- [出版物：“计算机与数学及其应用”，V39 N7-8（2000年4月）p253-261]
英文摘要：
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 < p depends on p), all nonzero pairsums of core residues are shown to be distinct, upto commutation. The known FLT case_1 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 units group G_k. -- Yielding main result: each residue mod p^k is the sum of at most four p-th power residues. Moreover, some results on the generative power (mod p^{k>2}) of divisors of p^2-1 are derived. -- [Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261]
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
