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

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, t of powerful generator p−1 = rs = tu of ±B k mod p k , and of p+1, are investigated as primitive root candidates. Fermat's Small Theorem: x p−1 ≡ 1 mod p for 0 < x < p is, by applying 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 ± 1 have distinct r n mod p 3 (0 < n ≤ 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: W ief erich primes [4] and F LT case 1. It is conjectured that at least one divisor of p ± 1 is a semi primitive root of 1 mod p k .