On the Solutions of the Exponential Diophantine Equation 3^x-n^y=2z^2
Article Sidebar
Main Article Content
Abstract
In this paper, we study the solutions of the exponential Diophantine equation , where
are non-negative integers and
is a positive integer. By using elementary properties of congruence and Leu and Li’s theorem, results are obtained that if
is even, then the equation has exactly four solutions, which are
, For some odd
, we show that if
, then the equation has non-negative integer solutions
in the form
,
,
and
, where
is a non-negative integer. This discovery provides the solutions in the case of
, namely, the solutions
of the equation in non-negative integers are given by
,
,
and
, where
is a non-negative integer. If
,
and the equation has a solution, then
is odd and
is even. If
,
and the equation has a solution, then
is even. Moreover, we prove that if
or
, then
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Journal of TCI is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
References
Rabago, J.F.T. (2013). On two Diophantine equations 2^x+3y^2=4^z and 2^x+7y^2=4^z. International Journal of Advanced Mathematical Sciences, 1(1), 23-25.
Rabago, J.F.T. (2018). On the Diophantine equation 4^x-p^y=3z^2 where p is a prime. Thai Journal of Mathematics, 16(3), 643-650.
Siraworakun, A., & Tadee, S. (2022). On two Diophantine equations 16^x+qp^y=z^4 and 16^x-qp^y=z^4. Rattanakosin Journal of Science and Technology, 4(3), 40-45. (in Thai)
Thongnak, S., Chuayjan, W., & Kaewong, T. (2022). On the exponential Diophantine equation 5^x-2∙3^y=z^2. Annals of Pure and Applied Mathematics, 25(2), 109-112.
Chuayjan, W., Thongnak, S., & Kaewong, T. (2023). On the exponential Diophantine equation 3^x-5^y=z^2. Annals of Pure and Applied Mathematics, 28(1), 25-28.
Tadee, S. (2023). On the Diophantine equation 3^x-p^y=z^2 where p is prime. Journal of Science and Technology Thonburi University, 7(1), 1-6.
Tadee, S. (2023). On the Diophantine equations (n+2)^x-2∙n^y=z^2 and (n+2)^x+2∙n^y=z^2. Annals of Pure and Applied Mathematics, 27(1), 19-22.
Tadee, S., & Pintoptang, U. (2023). On the non-linear Diophantine equations 4^x-a^y=dz^2 and 4^x+a^y=dz^2. Thai Journal of Mathematics, 21(3), 563-567.
Tadee, S. (2024). On the solutions of the Diophantine equations (p-1)^x+2∙p^y=z^2 and (p-1)^x-2∙p^y=z^2. Journal of Science and Technology CRRU, 3(2), 15-22. (in Thai)
Mavongsa, J., & Tadee, S. (2025). On the Diophantine equation 3^x-2∙5^y=z^2 . Journal of Science and Technology Phetchabun Rajabhat University, 5(2), 13-18. (in Thai)
Burton, D.M. 2010. Elementary number theory. 7th ed., New York: McGraw-Hill.
Leu, M., & Li, G. (2003). The Diophantine equation 2x^2+1=3^n. Proceedings of the American Mathematical Society, 131(12), 3643-3645.
