On primitive roots of unity, divisors of p+/-1, Wieferich primes, and quadratic analysis mod p^3

Abstract

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 Conj: at least one divisor of p \pm 1 is a semi primitive root of 1 mod p^k. -- (paper withdrawn, re thm2.2)