Every Sum of Two Positive Integers Has Either a Trivial or a Non-Trivial Common Factor
Joseph Wamukoya
Joseph Kongani Wamukoya, Department of Math/Physics, Nairobi, Westlands, Nairobi, Kenya.
Manuscript received on 14 November 2025 | First Revised Manuscript received on 24 November 2025 | Second Revised Manuscript received on 16 March 2026 | Manuscript Accepted on 15 April 2026 | Manuscript published on 30 April 2026 | PP: 1-2 | Volume-6 Issue-1, April 2026 | Retrieval Number: 100.1/ijam.A122806010426 | DOI: 10.54105/ijam.A1228.06010426
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© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: This paper investigates the arithmetic structure of exponential Diophantine equations of the form A x + B y= k n , where A, B, k, x, y, n ∈ Z + . Classical treatments such as the Beal Conjecture and Fermat’s Last Theorem (FLT) restrict attention to exponents greater than two, leaving open the structural behavior of the equation for n = 1 and n = 2. This manuscript provides a unified framework addressing all positive integer exponents. A central theorem establishes that each term A x, B y, and k n can be expressed as the sum of an arithmetic sequence whose number of terms and average term are positive integers, provided the equation has a trivial or non-trivial common factor. This elasticity property of k n is derived through Gauss’s method for summing arithmetic progressions. The case n = 2 recovers the classical identity for k 2 as the sum of the first k odd integers, revealing Pythagoras’ theorem as a special instance of the general framework. For exponents exceeding two, if gcd(A, B,k) = 1, the arithmetic structure collapses, aligning with the Beal Conjecture as it is presented in the literature as a generalization of FLT. The results demonstrate a consistent theory for all positive integer exponents and show that every sum of two positive integers has either a trivial or a non-trivial common factor.
Keywords: Beal Conjecture; Arithmetic Sequences; Exponential Diophantine Equations; Fermat’s Last Theorem; Pythagoras.
Scope of the Article: Number Theory