Abstract : The additive structure of multiplicative semigroup Z_k=Z(.) mod p^k (odd prime p) is analysed. Order (p-1)p^{k-1} of cyclic group G_k of units mod p^k implies product G_k = A_kB_k, with cyclic core |A_k| = p-1, and extension subgroup |B_k| = p^k-1 which consists of all units n = 1 mod p, generated by p+1. The p-th power residues n^p mod p^k in G_k form an order |G_k|/p subgroup F_k, with |F_k| / |A_k| = p^k-2, so F_k properly contains core A_k for k>2. The additive structure of subgroups G_k and F_k is derived by considering successor S(n)=n+1 and the two arithmetic symmetries C(n)=-n and I(n)=n^-1 as functions, with commuting IC = CI, but S not commuting with I and C, producing four distinct compositions SCI, CIS, CSI, ISC all having period 3 upon iteration. This yields a triplet structure in G_k of three inverse pairs (n_i, n_i^-1) with n_i+1 = -(n_i+1)^-1 for i=0,1,2 where n₀n₁n₂ = 1 mod p^k, generalizing the cubic root solution n+1 = - n^-1 = - n² mod p^k (p=1 mod 6). . . Any solution in core (x+y)^p = x+y = x^p + y^p mod p^k has the EDS property of exponent p distributing over a sum. This is employed to derive the known FLT inequality for integers: interpret in a case₁ FLT mod p^k equivalence the terms as pos.integers < p^k, with p-th powers < p^kp. Then the (p-1)k carries cause the FLT_1 inequality.

sgrp-flt.htm . . . (3 pgs) : Intro on function composition as the semigroup- extension of arithmetic.
scimat98.htm . . . Cubic roots of 1: breaking the Hensel lift, for the FLT inequality : the carry makes the difference.
carry.htm - Residue and Carry: Closure and Generation (p+1)^*.
FST mod p^3 . . . "Fermat's Small Theorem extended to r^{p-1} mod p^3 for divisors r | p \pm 1."
fermnote.pdf . . . "On Fermat's marginal note: a suggestion."

Amspade Research --- The Netherlands --- (c) 27 Dec.1999
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