On primitive roots of unity, divisors of p+/-1, Wieferich primes, and quadratic analysis mod p³

الأصل · EN

Primitive roots of 1 mod pᵏ (k>2 and odd prime p) are sought, in cyclic units group Gₖ = Aₖ Bₖ mod pᵏ, coprime to p, of order (p-1)pᵏ⁻¹. 'Core' subgroup Aₖ has order p-1 independent of k, and p+1 generates 'extension' subgroup Bₖ of all pᵏ⁻¹ residues 1 mod p. Divisors r,t of powerful generator p-1=rs=tu of ± Bₖ mod pᵏ, and of p+1, are investigated as primitive root candidates. Fermat's Small Theorem: xᵖ⁻¹ 1 mod p for 0<x<p is, with recursion rⁿ⁺¹-tⁿ⁺¹=(rⁿ-tⁿ)(r+t)-(rⁿ⁻¹-tⁿ⁻¹)rt (divisors r!= t) extended to: all divisors r | p ± 1 have distinct rⁿ mod p³ (0<n ≤ p). So for proper divisors: rᵖ⁻¹!= 1 mod p³, a necessary (not sufficient) condition for a primitive root mod pᵏ>². And for prime p: 2ᵖ!=2 and 3ᵖ!= 3 (mod p³). Re: Wieferich primes [4] and FLT case₁. Conj: at least one divisor of p ± 1 is a semi primitive root of 1 mod pᵏ. -- (paper withdrawn, re thm2.2)