Tasanai Rangpung, "On the Diophantine Equation $$a^x + (2a+12)^{y}=z^2$$ where $$a \equiv 8(mod 27)$$," WSEAS Transactions on Mathematics, vol. 24, pp. 750-755, 2025, DOI:10.37394/23206.2025.24.74
Tasanai Rangpung. On the Diophantine Equation $$a^x + (2a+12)^{y}=z^2$$ where $$a \equiv 8(mod 27)$$.
WSEAS Transactions on Mathematics. 2025;24:750-755. 10.37394/23206.2025.24.74
WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 24, 2025
On the Diophantine Equation $$a^x + (2a+12)^{y}=z^2$$ where $$a \equiv 8(mod 27)$$
Author:
Abstract: In this paper, we show that for all solutions of the Diophantine equation where $$x,y,z$$ are non-negative integers are present $$a$$ be a positive integer with $$a \equiv 8(mod 27)$$. It has exactly two non-negative integer infinite solutions $$(x,y,z)=(1,0,\sqrt{a+1})$$ where $$a=(9t\pm 3)^{2}-1$$ and $$(x,y,z)=(1,1,\sqrt{3a+12})$$ where $$a=243t^{2} \pm 108t + 8$$ . Moreover, we prove that $$(x,y,z)=(3,2,36)$$ is the unique non-negative integer solution and is an integer.
Keywords:
The Diophantine equation, non-negative integer, Congruence, infinite solutions, Mihăilescu’s Theorem, integer solutions
Pages: 750-755 WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 24, 2025, Art. #74 DOI: 10.37394/23206.2025.24.74
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