On the solutions of some Mersenne prime-involved Diophantine equation
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Date
2023-03-31
Journal Title
Journal ISSN
Volume Title
Publisher
International Journal of Mathematics and Computer Science
Abstract
This work studies Diophantine equations of the form Aˣ -Bʸ = Z². Specifically, we determine the nonnegative integer solutions (pM, a, b, c) of the exponential Diophantine equation (pM)ᵃ - (pM 1)ᵇ = c² and its more generalized form (pM)ᵃ -(pM 1) ᵇ = c²ⁿ, where pM is a Mersenne prime number. Moreover, we also deal with the Diophantine equation (pM)ᵃ - (qM 1)ᵇ = c², where pM and qM are both Mersenne primes. We solve these equations with the aid of elementary methods in number theory like the factoring technique and the modular arithmetic method. We also utilize Mihailescu's Theo- rem, the concepts of quadratic residue and Legendre symbol, and some properties of Mersenne primes for our Diophantine analysis. Results show that both (pM)ᵃ - (pM 1)ᵇ = c² and (pM)ᵃ - (pM 1)ᵇ = c²ⁿ have trivial solutions which only exist when a = 0 and b = 0, while (pM)ᵃ - (qM 1)ᵇ = c² has two positive integer solutions.
Description
Keywords
Diophantine equations, Mersenne primes, Number theory, Prime numbers, Exponential equations, Integer solutions, Mathematical proofs
Citation
Gayo, W. S., Jr., & Bacani, J. B. (2023). On the solutions of some Mersenne prime-involved Diophantine equation. International Journal of Mathematics and Computer Science, 18(3), 487-495. https://future-in-tech.net/18.3/R-MathTech22-Gayo-Bacani.pdf