On primitive roots of 1 mod p k, divisors of p ± 1, Wieferich primes, and quadratic analysis mod p 3

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p 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 precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p + 1. Integer divisors r, s of powerful generator p − 1 = rs of ±B k mod p k , and of p + 1, are investigated as primitive root candidates. Using (p − 1) p−1 ≡ p + 1 mod p 3 , Fermat's Small Theorem (F ST) : n p ≡ n mod p for 0 < n < p, is extended and complemented to: all divisors r | p ± 1 have distinct r p−1 mod p 3 , so r p ≡ / r mod p 3 for each proper divisor, a necessary (but not sufficient) condition for a primitive root mod p k>2. Hence 2 p ≡ / 2 mod p 3 for prime p > 2 (re: W ief erich primes [3] and F LT case 1) and 3 p ≡ / 3 mod p 3 for prime p > 3. It is conjectured that at least one divisor of p ± 1 is a semi primitive root of 1 mod p k .