Triplets as Additive Structure of the Units Group Mod and its Subgroup of -Th Powers, with an Integer Consequence
The additive structure of multiplicative semigroup mod (odd prime p) is analysed. Order ( 1)1 of cyclic group of units mod implies product , with cyclic 'core' , and 'extension subgroup' which consists of all units mod , generated by +1. The -th power residues mod in form an order subgroup , with , so properly contains core for ≥ 3.The additive structure of subgroups and is derived by considering successor () = + 1 and the two arithmetic symmetries () = and () = 1 as functions, with commuting = , but not commuting with and , producing four distinct compositions all having period 3 upon iteration. This yields a structure in of three inverse pairs with for = 0; 1; 2 where 1 mod , generalizing the cubic root solution mod ( = 1 mod 6).Any solution ' in core' has the property of exponent distributing over a sum, shown to imply the known inequality for integers. Interprete in an equivalence mod () the terms as naturals < , so -th powers < . Then the ( − 1) carries cause the inequality, and inequivalence mod +1 is derived for the cubic roots of 1 mod