Malaysian Journal of Mathematical Sciences, April 2019, Vol. 13(S)
Special Issue: 3rd International Conference on Mathematical Sciences and Statistics (ICMSS2018)
On the Diophantine Equation \(5^x+p^mn^y=z^2\)
Bakar, H. S., Sapar, S. H., and Johari, M. A. M.
Corresponding Email: sitihas@upm.edu.my
Received date: -
Accepted date: -
Abstract:
The 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 \((x,m,n,y,z)=(2r,t,p^tk^2\pm2k5^r,1,p^tk\pm5^r)\) and \(\left ( 2r,2t,\frac{5^{2r-\alpha}-5^\alpha}{2p^t},2,\frac{5^{2r-\alpha}+5^\alpha}{2} \right )\) for \(k,r,t \in \mathbb{N}\) where \(0\leq \alpha< r\).
Keywords: -
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