On the Diophantine Equation n^x-17^y=z^2 - Tadee, Suton (Burapha Science Journal, 2024)
Background and Objectives: to find the non-negative integer solutions of the Diophantine equation , where is a positive integer, which satisfies one of the following conditions: 1. 2. and 3. and . Methodology: to prove by using the basic concepts of number theory. Main Results: If , then the equation has the only non-negative integer solution . If , then the non-negative integer solutions are and <img title="\left ( x,y,z \right )=\left ( 1,t,\sqrt{n-17^{t}} \right )" src="https://latex.codecogs.com/gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;1,t,\sqrt{n-17^{t}}&space;\right&space;)" />, where is a non-negative integer such that is an integer. Moreover, if and , then the equation has the unique non-negative integer solution . Conclusions: Let be a positive integer. If or and , then the equation has the unique solution . If , then the equation has the solutions and <img title="\left ( x,y,z \right )=\left ( 1,t,\sqrt{n-17^{t}} \right )" src="https://latex.codecogs.com/gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;1,t,\sqrt{n-17^{t}}&space;\right&space;)" />, where is a non-negative integer such that is an integer.