| 关于不定方程5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3) |
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| 引用本文： | 杨晓柳,牟全武.关于不定方程5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3)[J].西南农业大学学报,2019,41(4):92-96. |
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| 作者姓名： | 杨晓柳 牟全武 |
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| 作者单位： | 西安工程大学 理学院, 西安 710048 |
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| 基金项目： | 国家自然科学基金项目（11271283）；西安工程大学基础课程质量提升项目（104020184）. |
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| 摘 要： | 运用Pell方程、递推序列、同余式及平方剩余等初等数论知识,证明了不定方程5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3)仅有4组非平凡整数解(x,y)=(6, 4),(-9, 4),(6,-7),(-9,-7),同时给出该不定方程的全部整数解,分别为(x,y)=(0, 0),(0,-1),(0,-2),(0,-3),(-1, 0),(-1,-1),(-1,-2),(-1,-3),(-2, 0),(-2,-1),(-2,-2),(-2,-3),(-3, 0),(-3,-1),(-3,-2),(-3,-3),(6, 4),(-9, 4),(6,-7),(-9,-7).
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| 关 键 词： | 不定方程 整数解 递推序列 平方剩余 |
| 收稿时间： | 2018/1/16 0:00:00 |
On the Diophantine Equation 5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3) |
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| YANG Xiao-liu,MU Quan-wu.On the Diophantine Equation 5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3)[J].Journal of Southwest Agricultural University,2019,41(4):92-96. |
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| Authors: | YANG Xiao-liu MU Quan-wu |
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| Institution: | School of Science, Xi''an Polytechnic University, Xi''an 710048, China |
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| Abstract: | Using some techniques of elementary number theory involving Pell''s equation, recurrent sequence, congruence and quadratic residues, this paper proves that the diophantine equation 5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3) has only four non-trivial solutions, i.e. (x, y)=(6, 4), (-9, 4), (6, -7), (-9, -7), and gives all of its integer solutions, i.e. (x, y)=(0, 0), (0, -1), (0, -2), (0, -3), (-1, 0), (-1, -1), (-1, -2), (-1, -3), (-2, 0), (-2, -1), (-2, -2), (-2, -3), (-3, 0), (-3, -1), (-3, -2), (-3, -3), (6, 4), (-9, 4), (6, -7), (-9, -7). |
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| Keywords: | diophantine equation integer solution recurrent sequence quadratic residue |
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