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Pinpointing the Unique Exponent Configuration for a Specific Exponential Diophantine Equation
Abstract
We established the complete solution set for a specific exponential Diophantine equation involving polynomial bases with three unknown exponents (k, m, n). Our analysis demonstrated that the equation possesses a unique positive integer solution for the exponents (k, m, n), namely (1, 1, 2), which holds true for every positive integer value of the parameter ’w’, subject to a set of arithmetic constraints. The proof is constructed through a rigorous synthesis
of established arithmetic methods (such as modular congruences and the Quadratic Reciprocity Law) and advanced analytic tools derived from transcendental number theory, specifically leveraging a powerful bound on p-adic measurement of logarithmic sums developed by Bugeaud. The proof’s numerical bounds and specific calculations were performed and confirmed using Python code.D61.
Keywords
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