# On the Diophantine Equations x^2+y^2+z^2=kxyz

 dc.contributor 洪有情 zh_TW dc.contributor Yu-Ching Hung en_US dc.contributor.author 林志穎 zh_TW dc.contributor.author Chih-Ying Lin en_US dc.date.accessioned 2019-09-05T01:13:20Z dc.date.available 2007-7-4 dc.date.available 2019-09-05T01:13:20Z dc.date.issued 2007 dc.description.abstract 這篇論文中，我們就k值來探討丟番圖方程x^2+y^2+z^2=kxyz之解的情形： (1)當k不為1和3時，此方程式無正整數解。 (2)當k=1時，有無限多組正整數解。若將解表為(a,b,c),a小於或等於b小 於或等於c,則 ① 當c=3p^n或c=6p^n時，有解必唯一。 ② 若c為奇數，當c-2=p^n或c+2=p^n時，有解必唯一。 ③ 若c為偶數，當c-2=4p^n或c+2=8p^n時，有解必唯一。 (3)當k=3時，即為大家熟知的馬可夫方程式。 zh_TW dc.description.abstract In this paper, we discuss the positive integers solutions of the Diophantine equations x^2+y^2+z^2=kxyz. (1)When k doesn't equal to 1 and 3, the equations have no positive integers solutions. (2)When k=1, the equation has infitely many positive integers solutions. We can let (a,b,c) be the solution and arrange its entries in ascending order. ①The solution is determined uniquely by c when c=3p^n or c=6p^n. ②If c is odd, the solution is determined uniquely by c when c-2=p^n or c+2=p^n . ③If c is even, the solution is determined uniquely by c when c-2=4p^n or c+2=8p^n. (3)When k=3, it is the well known Markoff equation. en_US dc.description.sponsorship 數學系 zh_TW dc.identifier GN0694400022 dc.identifier.uri http://etds.lib.ntnu.edu.tw/cgi-bin/gs32/gsweb.cgi?o=dstdcdr&s=id=%22GN0694400022%22.&%22.id.& dc.identifier.uri http://rportal.lib.ntnu.edu.tw:80/handle/20.500.12235/101736 dc.language 英文 dc.subject 馬可夫方程式 zh_TW dc.subject Markoff Equation en_US dc.title On the Diophantine Equations x^2+y^2+z^2=kxyz zh_TW

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