On the Diophantine Equation 5 x + p mn y = z 2

• Main Authors: Bakar, H. S., Sapar, S. H., Johari, M. A. M.
• Format: Article
• Language: English
• Published: Institute for Mathematical Research, Universiti Putra Malaysia 2019
• Online Access: http://psasir.upm.edu.my/id/eprint/81535/1/Diophantine.pdf; http://psasir.upm.edu.my/id/eprint/81535/; https://mjms.upm.edu.my/senaraimakalah.php?yr=2019&bln=April&vol=13(S)
• Institution: Universiti Putra Malaysia

Diophantine equation is a polynomial equation with two or more unknowns for which only integral solutions are sought. This paper concentrates on finding the integral solutions to the Diophantine equation 5 x + p mn y = z 2 where p > 5 a prime number and y = 1, 2. The positive integral solutions to the equation are (x, m, n, y, z) = (2r, t, pt k 2 ± 2k5 r , 1, pt k ± 5 r ) and 2r, 2t, 5 2r−α − 5 α 2p t , 2, 5 2r−α + 5α 2 for k, r, t ∈ N where 0 ≤ α < r.