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Ta có:x2−4xy+5y2=169⇔(x−2y)2+y=169" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">x2−4xy+5y2=169⇔(x−2y)2+y=169. Tuy nhiên, số 169" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">169 chỉ có thể viết dưới dạng tổng của hai số nguyên dương bình phương theo hai cách:169=02+132=52+122" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">169=02+132=52+122Từ đó ta có:{x−2y=0y=13" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=0y=13 hay {x−2y=13y=0" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=13y=0, hay{x−2y=0y=−13" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=0y=−13 hay {x−2y=−13y=0" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=−13y=0, hay{x−2y=5y=12" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=5y=12 hay {x−2y=12y=5" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=12y=5, hay{x−2y=−5y=12" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=−5y=12 hay {x−2y=12y=−5" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=12y=−5, hay{x−2y=5y=−12" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=5y=−12 hay {x−2y=−12y=5" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=−12y=5, hay{x−2y=−5y=−12" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=−5y=−12 hay {x−2y=−12y=−5" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">{x−2y=−12y=−5, haySuy ra phương trình đã cho có các nghiệm nguyên sau:(13,0);(−13,0);(26,13);(−26,−13);(29,12);" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">(13,0);(−13,0);(26,13);(−26,−13);(29,12);
(−29,−12);(19,12);(−19,−12);(22,5);(−22,−5)" role="presentation" style="font-size: 13.696px; display: inline; word-spacing: 0px; position: relative;">(−29,−12);(19,12);(−19,−12);(22,5);(−22,−5)
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