ADDITIVE STRUCTURE OF THE GROUP OF UNITS MOD p, WITH CORE AND CARRY CONCEPTS FOR EXTENSION TO INTEGERS

The additive structure of multiplicative semigroup Zpk = Z(·) mod pk is analysed for prime p > 2. Order (p − 1)pk−1 of cyclic group Gk of units mod pk implies product Gk ≡ AkBk , with cyclic ’core’ Ak of order p−1 so np ≡ n for core elements, and ’extension subgroup’ Bk of order p k−1 consisting of all units n ≡ 1 mod p, generated by p+1. The p-th power residues np mod pk in Gk form an order |Gk|/p subgroup Fk, with |Fk|/|Ak| = pk−2, so Fk properly contains core Ak for k ≥ 3. The additive structure of subgroups Ak, Fk and Gk is derived by successor function S(n) = n+1, and by considering the two arithmetic symmetries C(n) = −n and I(n) = n−1 as functions, with commuting IC = CI, where S does not commute with I nor C. The four distinct compositions SCI, CIS, CSI, ISC all have period 3 upon iteration. This yields a triplet structure in Gk of three inverse pairs (ni, n −1 i ) with ni + 1 ≡ −(ni+1)−1 for i = 0, 1, 2 where n0 · n1 · n2 ≡ 1 mod pk, generalizing the cubic root solution n + 1 ≡ −n−1 ≡ −n2 mod pk (p ≡ 1 mod 6). Any solution in core: (x + y)p ≡ x + y ≡ xp + yp mod pk>1 has exponent p distributing over a sum, shown to imply the known FLT inequality for integers. In such equivalence mod pk (FLT case1) the three terms can be interpreted as naturals n < pk, so np < pkp, and the (p − 1)k produced carries cause FLT inequality. In fact, inequivalence mod p3k+1 is derived for the cubic roots of 1 mod pk(p≡ 1 mod 6).