On the Diophantine equation 3^x+p^y.q^z=u^2
Keywords:
Diophantine equation, Prime number, Congruence, Legendre symbolAbstract
In this paper, we study and find all non-negative integer solutions of the Diophantine equation , where
and
are non-negative integers and
are prime numbers with
and
. By using mathematical proofs combined with basic knowledge of number theory, such as: prime number, congruence and Legendre symbol, the results of the research were found that the Diophantine equation has the unique non-negative integer solution. That is
. Moreover, it can be used to find the non-negative integer solutions of the Diophantine equation
, where
is a non-negative integer.
References
Sroysang, B. (2013). More on the Diophantine equation 2^x+3^y=z^2. International Journal of
Pure and Applied Mathematics, 84(2), p.p. 133-137.
Sroysang, B. (2013). On the Diophantine equation 3^x+17^y=z^2 . International Journal of Pure
and Applied Mathematics, 89(1), p.p. 111-114.
Rabago, J.F.T. (2013). On two Diophantine equations 3^x+19^y=z^2 and 3^x+91^y=z^2.
International Journal of Mathematics and Scientific Computing, 3(1), p.p. 28-29.
Asthana, S. and Singh, M.M. (2017). On the Diophantine equation 3^x+13^y=z^2 . International
Journal of Pure and Applied Mathematics, 114(2), p.p. 301-304.
Rao, C.G. (2018). On the Diophantine equation 3^x+7^y=z^2. EPRA International Journal of
Research and Development, 3(6), p.p. 93-95.
Asthana, S. and Singh, M.M. (2020). On the Diophantine equation 3^x+117^y=z^2 . GANITA,
(2), p.p. 43-47.
Tangjai, W. and Chubthaisong, C. (2021). On the Diophantine equation 3^x+p^y=z^2 where p=2(mod 3)
. WSEAS Transactions on Mathematics, 20, p.p. 283-287.
Manikandan, K. and Venkatraman, R. (2025). Exploring Diophantine equations Involving Twin
Primes: A note on 3^x+p^y=z^2 . Palestine Journal of Mathematics, 14(SII), p.p. 68-72.
Sroysang, B. (2014). On the Diophantine equation 3^x+45^y=z^2. International Journal of Pure
and Applied Mathematics, 91(2), p.p. 269-272.
Sroysang, B. (2014). More on the Diophantine equation 3^x+85^y=z^2. International Journal of
Pure and Applied Mathematics, 91(1), p.p. 131-134.
Sangam, B.R. (2020). On the Diophantine equations 3^x+6^y=z^2 and 5^x+8^y=z^2. Annals of
Pure and Applied Mathematics, 22(1), p.p. 7-11.
Biswas, D. (2022). Does the solution to the non-linear Diophantine equation 3^x+35^y=z^2 exists?. Journal of Scientific Research, 14(3), p.p. 861-865.
Srimud, K. and Tadee, S. (2023). On the Diophantine equation 3^x+b^y=z^2 . International
Journal of Mathematics and Computer Science, 18(1), p.p. 137-142.
Malavika, N. and Venkatraman, R. (2024). On the exponential Diophantine equation
^x+121^y=z^2. International Journal of Mathematics and Computer Science, 19(3), p.p. 917-
Biswas, D. (2025). A discussion on the solution(s) of the Diophantine equation 3^x+15^y=z^2 .
Journal of Scientific Research, 17(1), p.p. 129-132.
Ai-juan, X. and Mou-jie, D. (2015). On the Diophantine equation 3^x+5^y.19^z=u^2. Journal of
Progressive Research in Mathematics, 5(3), p.p. 578-581.
Vesarachasart, S., Makuwing, A., Wasurakka, N. and Phattharachittra, K. (2019). On the
Diophantine equation 3^x+5^y.7^z=u^2. Thai Journal of Science and Technology, 8(3), p.p.
-225.
Burton, D.M. (2010). Elementary number theory. 7th ed., New York: McGraw-Hill. p.p. 63-189.
Raksangoen, P., Tongnuy, P. and Tadee, S. (2025). On the solutions of the Diophantine
equation 7^x+35^y=z^2. Journal of Science and Technology RMUTSB, 9(1), p.p. 102-109.
Sroysang, B. (2012). On the Diophantine equation 3^ x+5^y=z^2. International Journal of Pure
and Applied Mathematics, 81(4), p.p. 605-608.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Journal of science and technology RMUTSB
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
