ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXIV, 2 (2005)
p. 169 - 184

Additive structure of the group of units mod pk, with core and carry concepts for extension to integers
N. F. Benschop

Abstract.  The additive structure of multiplicative semigroup Zpk = Z(.) mod pk is analysed for prime p > 2. Order (p 1)pk1 of cyclic group Gk of units mod pk implies product Gk º AkBk , with cyclic ’core’ Ak of order p1 so np º n for core elements, and ’extension subgroup’ Bk of order pk1 consisting of all units n º 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| = pk2, 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) = n1 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, ni1) with ni + 1 º -(ni+1)1 for i = 0,1,2 where n0 . n1 . n2 º 1 mod pk, generalizing the cubic root solution n + 1 º n1 º n2 mod pk (p º 1 mod 6).

Any solution in core: (x + y)p º x + y º 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(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.