On the solutions of some Mersenne prime-involved Diophantine equation

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creativework.keywordsExponential Diophantine equation, Mersenne prime, Mihailescu’s theorem.
dc.contributor.authorGayo, William S., Jr.
dc.contributor.authorBacani, Jerico B.
dc.date.accessioned2026-04-24T01:36:16Z
dc.date.available2026-04-24T01:36:16Z
dc.date.issued2023-03-31
dc.description.abstractThis 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.
dc.identifier.citationGayo, 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
dc.identifier.issn1814-0432
dc.identifier.urihttps://lakasa.dmmmsu.edu.ph/handle/123456789/1389
dc.language.isoen
dc.publisherInternational Journal of Mathematics and Computer Science
dc.relation.urihttps://future-in-tech.net/18.3/R-MathTech22-Gayo-Bacani.pdf
dc.sdgSDG 4
dc.sdgSDG 9
dc.subjectDiophantine equations
dc.subjectMersenne primes
dc.subjectNumber theory
dc.subjectPrime numbers
dc.subjectExponential equations
dc.subjectInteger solutions
dc.subjectMathematical proofs
dc.subject.ddcDiophantine equations
dc.subject.ddcNumber theory
dc.subject.lcshDiophantine equations
dc.subject.lcshNumber theory
dc.subject.lcshMersenne primes
dc.subject.lcshPrime numbers
dc.subject.lcshExponential equations
dc.subject.lcshMathematical analysis
dc.titleOn the solutions of some Mersenne prime-involved Diophantine equation
dc.typeArticle
dcterms.accessRightsOpen Access
oaire.citation.endPage495
oaire.citation.issue3
oaire.citation.startPage487
oaire.citation.volume18
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