Fermat’s Small Theorem Extended to r ^p−1mod p ³

Abstract

By the carries involved in (p±1)^p≡p ²±1 mod p ³, and by the lattice structure of Z(.)mod q for q=p±1 (odd prime p), all idempotents taken as naturals e<p are shown to have distinct e ^p−1 mod p ³, and divisors r of p−1 (resp. p+1) with different primesets have distinct r ^p−1 mod p ³. Moreover 2^p≢2  mod p ³ for prime p, related to Wieferich primes (Wieferich in J. Reine Angew. Math. 136:293–302, 1909) and FLT case₁ for integers (Chap. 8). Conjecture: Some g|p±1 is semi primitive root of 1 mod p ^k>2, with units group {−1, g}^*.