สมการไดโอแฟนไทน์ n^x-5^y=z^2  เมื่อ  n=11(mod20)

Suton Tadee

On the Diophantine Equation n^x-5^y=z^2 where n=11(mod20)

Authors

Suton Tadee Corresponding author

Keywords:

Diophantine equation, Non-negative integer solution

Abstract

In this paper, we show that $gif.latex?\left&space;(&space;x,y,z&space;\right&space;)=\left&space;(&space;0,0,0&space;\right&space;)$ is the unique non-negative integer solution of the Diophantine equation $gif.latex?n^{x}-5^{y}=z^{2}$ , where $gif.latex?n$ is a positive integer with $gif.latex?n\equiv&space;11\pmod{20}$ .

References

N. Burshtein, (2019). All the solutions of the Diophantine equations (p+1)^x -p^y=z^2 and p^y-(p+1)^x=z^2 when p is prime and x+y=2,3,4 . Annals of Pure and Applied Mathematics, 19(1), 53-57.

N. Burshtein, (2019). A short note on solutions of the Diophantine equations 6^x+11^y=z^2 and 6^x-11^y=z^2 in positive integers x,y,z . Annals of Pure and Applied Mathematics, 19(2), 55-56.

N. Burshtein, (2019). All the solutions of the Diophantine equations p^x+p^y=z^2 and p^x-p^y=z^2 when p is prime. Annals of Pure and Applied Mathematics, 19(2), 111-119.

N. Burshtein, (2020). All the solutions of the Diophantine equations 13^x-5^y=z^2 , 19^x-5^y=z^2 in positive integers x,y,z . Annals of Pure and Applied Mathematics, 22(2), 93-96.

A. Elshahed and H. Kamarulhaili, (2020). On the Diophantine equation (4^n)^x-p^y=z^2 . WSEAS Transactions on Mathematics, 19, 349-352.

W. Orosram and A. Unchai, (2022). On the Diophantine equation 2^{2nx}-p^y=z^2 , where p is a prime. International Journal of Mathematics and Computer Science, 17(1), 447-451.

S. Tadee and N. Laomalaw, (2022). On the Diophantine equations n^x-n^y=z^2 and 2^x-p^y=z^2. Phranakhon Rajabhat Research Journal (Science and Technology), 17(1), 10-16.

S. Thongnak, W. Chuayjan and T. Kaewong, (2019). On the exponential Diophantine equation 2^x-3^y=z^2 . Southeast-Asian Journal of Sciences, 7(1), 1-4.

S. Thongnak, W. Chuayjan and T. Kaewong, (2022). On the Diophantine equation 7^x-2^y=z^2 where x,y and z are non-negative integers. Annals of Pure and Applied Mathematics, 25(2), 63-66.