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 Title: Legendre的定理在Z[i]和Z[w]的情形;Legendre's Theorem in Z[i] and in Z[w] Authors: 施柏如;Shih,Po-Ju Contributors: 數學研究所 Keywords: Legendre's Theorem Date: 2004-01-16 Issue Date: 2014-05-08 15:26:30 (UTC+8) Publisher: 國立中央大學 Abstract: none;This thesis studies the Diophantine equation egin {eqnarray*}ax^{2}+by^{2}+cz^{2}=0,end {eqnarray*}which was investigated by Legendre when the coefficients are rational integers. Without loss of generality, we may assume that \$a,b,c\$ are nonzero integers, square free, and pairwise relatively prime. Legendre proved that the equation \$ax^{2}+by^{2}+cz^{2}=0\$ has a nontrivial integral solution if and only if egin{itemize}item[m (i)] \$a, b, c\$ are not of the same sign, anditem[m(ii)] \$-bc, -ac,\$ and \$-ab\$ are quadratic residues of \$a,b,\$ and \$c\$ respectively.end{itemize}The purpose of this thesis is to extend Legendre's Theorem by carrying over the cases withthe coefficients and unknowns in \${mathbb Z}[i]\$ and in \${mathbb Z}[omega]\$,where \$i\$ is a square root of \$-1\$ and \$omega\$ is a cubic root of unity.More precisely, we show that the necessary and sufficient conditions for the Diophantine equation \$ax^{2}+by^{2}+cz^{2}=0\$having a nontrivial solution over \${mathbb Z}[i]\$ is that \$bc, ca,ab\$ are quadratic residues mod \$a,b,c\$ respectively,and the equation having a nontrivial solution over \${mathbb Z}[omega]\$ is that \$-bc, -ca, -ab\$ are quadratic residuesmod \$a,b,c\$ respectively. Appears in Collections: [數學研究所] 博碩士論文

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