M ar 2 00 1 On primitive roots of unity , divisors of p 2 − 1 , and an extension to mod p 3 of Fermat ’ s Small Theorem

On primitive roots of unity, divisors of p 2 − 1, and an extension to 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 of powerful generators p±1 of ±B k mod p k are investigated, as primitive root candidates. Fermat's Small Theorem (F ST): all n < p have n p ≡ n mod p is extended, or rather complemented, to: all proper divisors r | p 2 − 1 have r p ≡ / r mod p 3 , a necessary (although not sufficient) condition for a primitive root mod p k>2. Hence 2 p ≡ / 2 mod p 3 for primes p > 2 (re: W ief erich primes [3] and F LT case 1), and 3 p ≡ / 3 mod p 3 for p > 3. It is conjectured that at least one divisor of p 2 − 1 is a semi primitive root of 1 mod p k (k ≥ 3).