Solution of the exponential Diophantine Equation

Vipawadee Moonchaisook

Solution of the exponential Diophantine Equation n^x+(2p-1)^y=z^2

ผู้แต่ง

Vipawadee Moonchaisook Department of Mathematics, Faculty of Science and Technology Surindra Rajabhat University

คำสำคัญ:

Diophantine equations, congruence, integer solutions, number theory

บทคัดย่อ

This study investigates the exponential Diophantine equation $n^{x}+\left(2p-1\right)^{y}=z^{2}$ where p is a prime number and n, x, y, z are non-negative integers, subject to the modular condition

$n 5(mod 12)with gcd\left(n,2p-1\right)=1.$

The primary objective is to determine all non-negative integer solutions of this equation by employing quadratic residue theory, modular arithmetic, and its connections to Pell-type equations.

The results demonstrate that the equation admits a unique non-negative integer solution given by

$\left(n,p,x,y,z\right)=\left(n,2,0,1,2\right)$

For all other values of p, no non-negative integer solutions exist, and it can be rigorously proven that cannot be a perfect square outside this solution. These findings provide a clear classification of the solution set structure and offer theoretical insights beneficial for further studies on exponential Diophantine equations, including potential applications in computational number theory and cryptographic systems.

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ดาวน์โหลด

เผยแพร่แล้ว

2025-11-04

รูปแบบการอ้างอิง

Moonchaisook, V. (2025). Solution of the exponential Diophantine Equation n^x+(2p-1)^y=z^2. วารสารวิทยาศาสตร์และเทคโนโลยี หัวเฉียวเฉลิมพระเกียรติ, 11(2), 49–58. สืบค้น จาก https://ph02.tci-thaijo.org/index.php/scihcu/article/view/259418