关于单位的原根、p+/-1的因子、Wieferich素数和模p^3的二次分析
On primitive roots of unity, divisors of p+/-1, Wieferich primes, and quadratic analysis mod p^3
摘要：
在与p互质的（p-1）p^{k-1}阶循环单位群G_k = A_kB_kmodp ^k中，求1 mod p ^k（k>2，奇素数p）的原根.“核心”子群A_k具有与k无关的p-1阶，p+1生成所有p^{k-1}剩余1 mod p的“扩张”子群B_k。\pm B_k mod p^k和p+1的强生成元p-1=rs=tu的除数r，t被研究为原根候选。费马小定理：x^{p-1} \e 1 mod p（0<x<p）是，递归r^{n +1}-t^{n +1}=（r^n-t^n）（r+t）-（r^{n-1}-t^{n-1}）rt（除数r！= t）扩展到：所有除数r| p \pm 1具有不同的r^n mod p^3（0<n \leq p）。所以对于真因子：r^{p-1}！= 1 mod p^3，原根mod p^{k>2}的必要（非充分）条件。对于素数p：2^p！= 2和3 ^p！= 3（mod p^3）。Re：Wieferich primes [4] and FLT case_1. Conj：p \pm 1的至少一个除数是1 mod p^k的半原根。--（撤回论文，回复thm2.2）
英文摘要：
Primitive roots of 1 mod p^k (k>2 and odd prime p) are sought, in cyclic units group G_k = A_k B_k mod p^k, coprime to p, of order (p-1)p^{k-1}. 'Core' subgroup A_k has order p-1 independent of k, and p+1 generates 'extension' subgroup B_k of all p^{k-1} residues 1 mod p. Divisors r,t of powerful generator p-1=rs=tu of \pm B_k mod p^k, and of p+1, are investigated as primitive root candidates. Fermat's Small Theorem: x^{p-1} \e 1 mod p for 0<x<p is, with recursion r^{n+1}-t^{n+1}=(r^n-t^n)(r+t)-(r^{n-1}-t^{n-1})rt (divisors r != t) extended to: all divisors r | p \pm 1 have distinct r^n mod p^3 (0<n \leq p). So for proper divisors: r^{p-1} != 1 mod p^3, a necessary (not sufficient) condition for a primitive root mod p^{k>2}. And for prime p: 2^p !=2 and 3^p != 3 (mod p^3). Re: Wieferich primes [4] and FLT case_1. Conj: at least one divisor of p \pm 1 is a semi primitive root of 1 mod p^k. -- (paper withdrawn, re thm2.2)
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