สมการไดโอแฟนไทน์ p2x + qy = z4 และ p2x – qy – z4 เมื่อ p และ q เป็นจำนวนเฉพาะ
On the Diophantine Equations p2x + qy = z4 and p2x – qy – z4 where p and q are Primes
Keywords:
สมการไดโอแฟนไทน์, ข้อคาดการณ์ของกาตาลัน, Diophantine equation , Catalan’s ConjectureAbstract
ในงานวิจัยนี้ได้ศึกษาสมการไดโอแฟนไทน์ p2x + qy = z4 และ p2x – qy – z4 เมื่อ p และ q เป็นจำนวนเฉพาะ พบว่า สมการไดโอแฟนไทน์ p2x + qy = z4 มีผลเฉลยทั้งหมดที่เป็นจำนวนเต็มที่ไม่เป็นลบคือ (p,q,x,y,z) ∈ {(3,7,1,1,2)} È {(p,2,1,log2 (p+1) + 2, |log2 (p+1), } È {(2,17,3,1,3)} และสมการไดโอแฟนไทน์ p2x – qy = z4 มีผลเฉลยทั้งหมดที่เป็นจำนวนเต็มที่ไม่เป็นลบคือ (p,q,x,y,z) ∈ {(p,q,1,logq (2p-1), |log2 (2p-1), } È {(p,2,1,log2 (p-1) + 2, |log2 (p-1), } È {(p,q,0,0,0)} È {(p,p,u,2,u,0) |u+ } In this paper, we study Diophantine equations p2x + qy = z4 and p2x – qy – z4 where p and q are primes. We found that all non-negative integer solutions of the Diophantine equation p2x + qy = z4 are of the following (p,q,x,y,z) ∈ {(3,7,1,1,2)} È {(p,2,1,log2 (p+1) + 2, |log2 (p+1), } È {(2,17,3,1,3)} and all non-negative integer solutions of the Diophantine equation p2x - qy = z4 are of the following (p,q,x,y,z) ∈ {(p,q,1,logq (2p-1), |log2 (2p-1), } È {(p,2,1,log2 (p-1) + 2, |log2 (p-1), } È {(p,q,0,0,0)} È {(p,p,u,2,u,0) |u+ }References
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