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

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

2001-03-12

arxiv.org/abs/math/0103067v10

Mathematics

General Mathematics

(5 pgs) Ref N.F.B: "Powersums representing residues mod p^k, from Fermat to Waring", Computers & Math's with Applic's, V39 N7-

Scientific paper

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 Conj: at least one divisor of p \pm 1 is a semi primitive root of 1 mod p^k. -- (paper withdrawn, re thm2.2)

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