สมการไดโอแฟนไทน์   n^x-17^y=z^2

Suton Tadee

Authors

Suton Tadee Faculty of Science and Technology, Thepsatri Rajabhat University

Keywords:

Diophantine equation , non-negative integer solution , congruence

Abstract

Background and Objectives: to find the non-negative integer solutions $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)$ of the Diophantine equation $gif.latex?n^{x}-17^{y}=z^{2}$ , where $gif.latex?n$ is a positive integer, which satisfies one of the following conditions: 1. $gif.latex?n\equiv&space;0\left&space;(&space;mod&space;4\right&space;)$ 2. $gif.latex?n\equiv&space;2\left&space;(&space;mod&space;4\right&space;)$ and 3. $gif.latex?n\equiv&space;3\left&space;(&space;mod&space;4\right&space;)$ and $gif.latex?n\equiv&space;\pm&space;1,\pm&space;2,\pm&space;4,\pm&space;8\left&space;(&space;mod&space;17\right&space;)$ .

Methodology: to prove by using the basic concepts of number theory.

Main Results: If $gif.latex?n\equiv&space;0\left&space;(&space;mod&space;4\right&space;)$ , then the equation has the only non-negative integer solution $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;0,0,0&space;\right&space;)$ . If $gif.latex?n\equiv&space;2\left&space;(&space;mod&space;4\right&space;)$ , then the non-negative integer solutions are $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;0,0,0&space;\right&space;)$ and $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;1,t,\sqrt{n-17^{t}}&space;\right&space;)$ , where $gif.latex?t$ is a non-negative integer such that $gif.latex?\sqrt{n-17^{t}}$ is an integer. Moreover, if $gif.latex?n\equiv&space;3\left&space;(&space;mod&space;4\right&space;)$ and $gif.latex?n\equiv&space;\pm&space;1,\pm&space;2,\pm&space;4,\pm&space;8\left&space;(&space;mod&space;17\right&space;)$ , then the equation has the unique non-negative integer solution $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;0,0,0&space;\right&space;)$ .

Conclusions: Let $gif.latex?n$ be a positive integer. If $gif.latex?n\equiv&space;0\left&space;(&space;mod&space;4\right&space;)$ or $gif.latex?n\equiv&space;3\left&space;(&space;mod&space;4\right&space;)$ and $gif.latex?n\equiv&space;\pm&space;1,\pm&space;2,\pm&space;4,\pm&space;8\left&space;(&space;mod&space;17\right&space;)$ , then the equation has the unique solution $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;0,0,0&space;\right&space;)$ . If $gif.latex?n\equiv&space;2\left&space;(&space;mod&space;4\right&space;)$ , then the equation has the solutions $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;0,0,0&space;\right&space;)$ and $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;1,t,\sqrt{n-17^{t}}&space;\right&space;)$ , where $gif.latex?t$ is a non-negative integer such that $gif.latex?\sqrt{n-17^{t}}$ is an integer.

References

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On the Diophantine Equation n^x-17^y=z^2

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