Additive Structure of Units Group mod p ^k, with Carry Extension for a Proof of Fermat’s Last Theorem

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 ≡A _k B _k , with cyclic ‘core’ A _k of order p−1 so n ^p≡n for core elements, and ‘extension subgroup’ B _k of order p ^k−1 consisting 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≥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 ⁻¹ _i ) with n _i +1≡−(n _i+1)⁻¹ for i=0,1,2 where n ₀.n ₁.n ₂≡1 mod p ^k, generalizing the cubic root solution n+1≡−n ⁻¹≡−n ² mod p ^k (p≡1 mod 6). Any solution in core: (x+y)^p≡x+y≡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 case 1) 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. Inequivalence mod p ^3k+1 is shown for the cubic roots of 1mod p ^k (p≡1mod 6).

(c) 2005 Bratislava University Press, with permission taken from [10].