ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
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 p 1 so np 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| = 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). ©
Copyright 2005, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
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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Institute of Applied Mathematics
Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic
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