NONE 20260206065543125 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Tasanai Rangpung In this paper, we show that for all solutions of the Diophantine equation where x,y,z are nonnegative integers are present a be a positive integer with a≡8(mod27). It has exactly two non-negative integer infinite solutions (x,y,z)=(1,0,√(a+1)) where a=(9t±3)^2-1 and (x,y,z)=(1,1,√(3a+12)) where a=243t^2±108t+8. Moreover, we prove that (x,y,z)=(3,2,36) is the unique non-negative integer solution and t is an integer. 12 31 2025 12 31 2025 750 74 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2025.24.74 https://wseas.com/journals/mathematics/2025/b505106-2184.pdf Elshahed and Kamarulhaili, On the Diophantine equation (4^n )^x-p^y=z^2, WSEAS transaction on Mathematics, Vol.19, 2020, pp. 349-352. https://doi.org/10.37394/23206.2020.19.35. Blinn, How to solve a cubic equation. Part 1. The shape of the discriminant, IEEE Computer Graphics and Applications, Vol.19, 2006, pp. 84-93. https://doi.org/10.1109/MCG.2006.60. Orosram and Comemuang, On the Diophantine equation 8^x+n^y=z^2, WSEAS transaction on Mathematics, Vol.19, 2020, pp. 520-522. https://doi.org/10.37394/23206.2020.19.56. Viriyapong, On the Diophantine Equation nx +13y=z2 where n=2(mod39) and n+1 is not a Square Number, WSEAS Transaction on Mathematics, Vol.20, 2021, pp. 442-445. https://doi.org/10.37394/23206.2021.20.45. Tangjai and Chubthaisong, On the Diophantine equation 3^x+p^y=z^2 where p≡2(mod3), WSEAS Transaction on Mathematics, Vol.22, 2021, pp. 283-287. https://doi.org/10.37394/23206.2021.20.29. Thongnak, Chuayjan and Kaewong, On the Exponential Diophantine equation 5^x-2⋅3^y=z^2, Annal of Pure and Applied Mathematics, Vol.25, No.2, 2022, pp. 109112. Tadee, On the Diophantine equation n^x+〖10〗^y=z^2, WSEAS Transaction on Mathematics, Vol.22, 2023, pp. 150-153. https://doi.org/10.37394/23206.2023.22.19. Srimud and Tadee, On the Diophantine equation 3^x+b^y=z^2, International Journal of Mathematics and Computer Science, Vol.18, 2023, pp. 137-172. Rangpung, Infinite solution of the Diophantine equation a^x+(2a)^y=z^2 where a≡3(mod16), International Journal of Mathematics and Computer Science, Vol.20, 2025, pp. 805-809. Mihăilescu, Primary cyclotomic units and a proof of Catalan's conjecture, Journal für die reine and angewandte Mathematik, Vol.572, 2004, pp. 167-195.