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[2] viXra:2603.0070 [pdf] submitted on 2026-03-12 19:42:46
Authors: Kurmet Sultan
Comments: 3 Pages.
We show that for any Pythagorean triple (a, b, c) satisfying a^2+b^2 = c^2 with a, b, c ∈ N, the equality c^3 −b^3 = a^3 cannot hold. The proof is elementary and relies only on algebraic manipulation and the structure of the Pythagorean equation, without invoking Fermat’s Last Theorem.
Category: Algebra
[1] viXra:2603.0060 [pdf] submitted on 2026-03-11 20:31:25
Authors: Youssef Ayyad
Comments: 35 Pages. (Header blocked by viXra Admin)
This article presents a systematic exposition of the duality between two complementary geometric characterizations of prime numbers, united through Pontryagin duality for finite cyclic groups. The first characterization is internal: an integer n ≥ 2 is prime if and only if everyequitable partition of the vertices of the regular n-gon Pn is fixed classwise only bythe identity rotation. We establish a canonical bijection between equitable partitionsof Pn and subgroups of the cyclic group Cn ∼= Z/nZ (Theorem 3.6). The second characterization is external and fractal: an integer n ≥ 2 is prime if and only if the regular polygon Gn is a primitive leaf in the universal subdivisiontree—that is, Gn has no ancestors other than the root G1. We prove that Gn appears in the fractal hierarchy F(m) generated by Gm if and only if m | n (Lemma 4.4), establishing a bijection between fractal ancestors of Gn and quotients of Cn (Theorem 4.8). These two characterizations are not merely equivalent reformulations of the arithmetic definition of primality; they are canonically dual. The duality is mediated bythe Pontryagin isomorphism Φ : Sub(Cn) → Quot(Cn) sending a subgroup H ⊆ Cn to its quotient Cn/H. For each proper divisor d | n with 1 < d < n, this duality pairs:u2022 Internally: the subgroup Hn/d ⊆ Cn of order n/d, realized geometrically as an equitable partition Pd of Pn into d classes; u2022 Externally: the quotient Cn/Hn/d ∼= Cd, realized geometrically as the ancestor polygon Gd in the fractal tree. These correspondences fit into a commutative diagram (Theorem 5.1) that unifies the two geometric visions: the same divisor d manifests simultaneously as a symmetriccoloring of the polygon (internal witness) and as a smaller polygon from which Gn descends (external witness). We develop this duality, with complete proofs, extensive examples, and visualizations. The framework then generalizes naturally: for arbitrary finite groups, rings,and modules, the duality between substructures and quotient structures can be rendered visible through dual geometric realizations. This suggests a broad research program in dual geometric algebra, where abstract algebraic simplicity—the absence of non-trivial substructures or quotients—acquires concrete geometric meaning.
Category: Algebra
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