Browsing by Author "Gayo, William S., Jr."

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    Analytic Geometry and Calculus (MATH 103)
    (Don Mariano Marcos Memorial State University - North La Union Campus, 2020) Gayo, William S., Jr.
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    On the Diophantine Equation Mxp + (Mq + 1)y = z2
    (European Journal of Pure and Applied Mathematics, 2021) Gayo, William S., Jr.; Bacani, Jerico B.
    In this paper, we study and solve the exponential Diophantine equation of the form Mxp + (Mq + 1)y = z2 for Mersenne primes Mp and Mq and non-negative integers x, y, and z. We use elementary methods, such as the factoring method and the modular arithmetic method, to prove our research results. Several illustrations are presented, as well as cases where solutions to the Diophantine equation do not exist.
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    On the solutions of some Mersenne prime-involved Diophantine equation
    (International Journal of Mathematics and Computer Science, 2023-03-31) Gayo, William S., Jr.; Bacani, Jerico B.
    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.