Triplets and Symmetries of Arithmetic Mod P
نویسنده
چکیده
The finite ring Zk = Z(+, .) mod p k of residue arithmetic with odd prime power modulus is analysed. The cyclic group of units Gk in Zk(.) has order (p − 1).p , implying product structure Gk ≡ Ak.Bk with |Ak| = p− 1 and |Bk| = p , the ”core” and ”extension subgroup” of Gk respectively. It is shown that each subgroup S ⊃ 1 of core Ak has zero sum, and p+1 generates subgroup Bk of all n ≡ 1 mod p in Gk. The p-th power residues n p mod p in Gk form an order |Gk|/p subgroup Fk, with |Fk|/|Ak| = p , so Fk properly contains core Ak for k ≥ 3. By quadratic analysis (mod p) rather than linear analysis (mod p, re: Hensel’s lemma [5] ), the additive structure of subgroups Gk and Fk is derived. Successor function n+1 combines with the two arithmetic symmetries −n and n to yield a triplet structure in Gk of three inverse pairs (ni, n −1 i ) with: ni + 1 ≡ −(ni+1) , indices mod 3, and n0.n1.n2 ≡ 1 mod p. In case n0 ≡ n1 ≡ n2 ≡ n this reduces to the cubic root solution n+ 1 ≡ −n −1 ≡ −n mod p (p=1 mod 6). . . The property of exponent p distributing over a sum of core residues : (x+y) ≡ x+y ≡ x +y mod p is employed to derive the known FLT inequality for integers. In other words, to a FLT mod p equivalence for k digits correspond p-th power integers of pk digits, and the (p− 1)k carries make the difference, representing the sum of mixed-terms in the binomial expansion.
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