สมการไดโอแฟนไทน์ p^{2x}+q^{2y}=z^2 และ p^{2x}-q^{2y}=z^2

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เผยแพร่แล้ว: ส.ค. 30, 2022
คำสำคัญ:
Diophantine equation, integer solution, Catalan’s conjecture

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Suton Tadee
Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, Lopburi 15000

บทคัดย่อ

ให้ gif.latex?p,q เป็นจำนวนเฉพาะ ในงานวิจัยนี้แสดงว่าสมการไดโอแฟนไทน์ gif.latex?p^{2x}+q^{2y}=z^{2} มีผลเฉลยที่เป็นจำนวนเต็มบวกเพียงผลเฉลยเดียว คือ gif.latex?\left&space;(&space;p,q,x,y,z&space;\right&space;)=\left&space;(&space;3,2,1,2,5&space;\right&space;) และผลเฉลยทั้งหมดที่เป็นจำนวนเต็มบวกของสมการไดโอแฟนไทน์ gif.latex?p^{2x}-q^{2y}=z^{2} อยู่ในรูป gif.latex?\left&space;(&space;p,q,x,y,z&space;\right&space;)=\left&space;(4^{n-1}+1,2,1,n,4^{n-1}-1&space;\right&space;) เมื่อ gif.latex?n\neq&space;1 เป็นจำนวนเต็มบวก

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ฉบับ
ปีที่ 30 ฉบับที่ 4 (กรกฎาคม-สิงหาคม 2565)
บท
Physical Sciences

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