Geodesic
Additive structure of the group of units mod p k , with core and carry concepts for extension to integers
Acta mathematica Universitatis Comenianae, Tome 74 (2005) no. 2 
N. F. Benschop. Additive structure of the group
of units mod p k ,
with core and carry concepts for extension to integers. Acta mathematica Universitatis Comenianae, Tome 74 (2005) no. 2. http://geodesic.mathdoc.fr/item/AMUC_2005_74_2_a2/
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     year = {2005},
     volume = {74},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2005_74_2_a2/}
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The additive structure of multiplicative semigroup Z pk = 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 k B k , with cyclic ’core’ A k of order p – 1 so n p o n for core elements, and ’extension subgroup’ Bk of order pk –1 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| = 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, ni–1) 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 –n–1 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. Download: Adobe PDF Compressed Postscript Version to read: Adobe PDF Acta Mathematica Universitatis Comenianae Institute of Applied Mathematics Faculty of Mathematics, Physics and Informatics Comenius University 842 48 Bratislava, Slovak Republic Telephone: + 421-2-60295755 Fax: + 421-2-65425882 e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc © Copyright 2005, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE