ผลเฉลยของสองสมการไดโอแฟนไทน์ n^{2x}+2^y=z^2 และ n^{2x}-2^y=z^2

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จันทนา วรรณพันธุ์
สุธน ตาดี

บทคัดย่อ

ในงานวิจัยนี้ แสดงว่า ผลเฉลยทั้งหมดที่เป็นจำนวนเต็มที่ไม่เป็นลบของสมการไดโอแฟนไทน์ gif.latex?n^{2x}+2^{y}=z^{2}


เมื่อ gif.latex?n เป็นจำนวนเต็มบวกคี่ อยู่ในรูปgif.latex?\left&space;(&space;n,x,y,z&space;\right&space;)\in&space;\left&space;\{&space;\left&space;(&space;1,a,3,3&space;\right&space;):a\in&space;\mathbb{Z},a\geq&space;0&space;\right&space;\}\cup&space;\left&space;\{&space;\left&space;(&space;b,0,3,3&space;\right&space;):b\in&space;\mathbb{Z},b>1&space;\right&space;\}\cupgif.latex?\left&space;\{&space;\left&space;(&space;2^{c-2}-1,1,c,2^{c-2}+1&space;\right&space;):c\in&space;\mathbb{Z},c>3&space;\right&space;\}


และผลเฉลยทั้งหมดที่เป็นจำนวนเต็มที่ไม่เป็นลบของสมการไดโอแฟนไทน์ gif.latex?n^{2x}-2^{y}=z^{2}เมื่อ gif.latex?n เป็นจำนวนเต็มบวกคี่


อยู่ในรูปgif.latex?\left&space;(&space;n,x,y,z&space;\right&space;)\in&space;\left&space;\{&space;\left&space;(&space;1,d,0,0&space;\right&space;):d\in&space;\mathbb{Z},d\geq&space;0&space;\right&space;\}\cup&space;\left&space;\{&space;\left&space;(&space;e,0,0,0&space;\right&space;):e\in&space;\mathbb{Z},e>1&space;\right&space;\}\cupgif.latex?\left&space;\{&space;\left&space;(&space;2^{f-2}+1,1,f,2^{f-2}-1&space;\right&space;):f\in&space;\mathbb{Z},f>3&space;\right&space;\}\cup&space;\left&space;\{&space;\left&space;(&space;3,1,3,1&space;\right&space;),\left&space;(&space;3,2,5,7&space;\right&space;)&space;\right&space;\}

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How to Cite
วรรณพันธุ์ จ., & ตาดี ส. (2022). ผลเฉลยของสองสมการไดโอแฟนไทน์ n^{2x}+2^y=z^2 และ n^{2x}-2^y=z^2. วารสารวิทยาศาสตร์และวิทยาศาสตร์ศึกษา (JSSE), 5(2), 236–240. https://doi.org/10.14456/jsse.2022.27 (Original work published 6 สิงหาคม 2022)
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Qi, L. and Li, X. (2015). The Diophantine equation〖 8〗^x+p^y=z^2. The Scientific World Journal, Article ID 306590, 3 pages. doi: 10.1155/2015/306590

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Sroysang, B. (2013). More on the Diophantine equation〖 2〗^x+3^y=z^2. International Journal of Pure and Applied Mathematics, 84(2), 133-137. doi: 10.12732/ijpam.v84i2.11

Suvarnamani, A., Singta, A. and Chotchaisthit, S. (2011). On two Diophantine equations 〖 4〗^x+7^y=z^2 and 〖 4〗^x+11^y=z^2. Science and Technology RMUTT Journal, 1(1), 25-28.

Tanakan, S. (2014). On the Diophantine equation〖 19〗^x+2^y=z^2. International Journal of Contemporary Mathematical Sciences, 9(4), 159-162. doi: 10.12988/ijcms.2014.418

Chotchaisthit, S. (2012). On the Diophantine equation 4^x+p^y=z^2 where p is a prime number. American. Jr. of Mathematics and Sciences, 1(1), 191-193.

Chotchaisthit, S. (2013). On the Diophantine equation 2^x+11^y=z^2. Maejo International Journal of Science and Technology, 7(2), 291-293.

Dhurga, C.K. (2021). A linear Diophantine equation and its real life applications. Advances and Applications in Mathematical Sciences, 20(8), 1389-1394.

Kaur, D. & Sambhor, M. (2017). Diophantine equations and its applications in real life. International Journal of Mathematics and its applications, 5(2), 217-222.

Khan, M.A.A., Rashid, A. & Uddin, M.S. (2016). Non-negative integer solutions of two Diophantine equations 2^x+9^y=z^2 and 5^x+9^y=z^2. Journal of Applied Mathematics and Physics, 4, 762-765. doi: 10.4236//jamp.2016.44086

Mihailescu, P. (2004). Primary cyclotomic units and a proof of Catalan’s conjecture. Journal für die Reine und Angewante Mathematik, 27, 167-195.

Puangjumpa, P. (2016). Possible solution of the Diophantine equation 2^x+47^y=z^2. Academic Journal URU, 11(3S), 36-42.

Qi, L. & Li, X. (2015). The Diophantine equation 8^x+p^y=z^2. The Scientific World Journal, Article ID 306590, 3 pages. doi: 10.1155/2015/306590

Rabago, J.F.T. (2016). On the Diophantine equation 2^x+17^y=z^2. Journal of the Indonesian Mathematical Society, 22(2), 177-181.

Sroysang, B. (2013). More on the Diophantine equation 2^x+3^y=z^2. International Journal of Pure and Applied Mathematics, 84(2), 133-137. doi: 10.12732/ijpam.v84i2.11

Suvarnamani, A., Singta, A., & Chotchaisthit, S. (2011). On two Diophantine equations 4^x+7^y=z^2 and 4^x+11^y=z^2. Science and Technology RMUTT Journal, 1(1), 25-28.

Tanakan, S. (2014). On the Diophantine equation 19^x+2^y=z^2. International Journal of Contemporary Mathematical Sciences, 9(4), 159-162. doi: 10.12988/ijcms.2014.418