Abstract.  The additive structure of multiplicative semigroup Z_p^k = Z(^.) mod p^k is analysed for prime p > 2. Order (p – 1)p^k–1 of cyclic group G_k of units mod p^k implies product G_k º A_kB_k , with cyclic ’core’ A_k of order p – 1 so n^p º n for core elements, and ’extension subgroup’ B_k of order p^{k –1} consisting of all units n º 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 > 3.

The additive structure of subgroups A_k, F_k and G_k 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 G_k of three inverse pairs (n_i, n_i^–1) with n_i + 1 º -(n_i+1)^–1 for i = 0,1,2 where n₀ ^. n₁ ^. n₂ º 1 mod p^k, generalizing the cubic root solution n + 1 º –n^–1 º –n² mod p^k (p º 1 mod 6).

Any solution in core: (x + y)^p º x + y º x^p + y^p mod p^k>1 has exponent p distributing over a sum, shown to imply the known FLT inequality for integers. In such equivalence mod p^k (FLT case₁) the three terms can be interpreted as naturals n < p^k, so n^p < p^kp, and the (p – 1)k produced carries cause FLT inequality. In fact, inequivalence mod p^3k+1 is derived for the cubic roots of 1 mod p^k(pº 1 mod 6).

Keywords: Residue arithmetic, ring, group of units, multiplicative semigroup, additive structure, triplet, cubic roots of unity, carry, Hensel, Fermat, FST, FLT  

AMS Subject classification:  11D41, 11P99, 11A15.

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