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].西南大学学报(自然科学版),2019,41(4):92~96
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关于不定方程5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3)
杨晓柳, 牟全武
西安工程大学 理学院, 西安 710048
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5xx+1)(x+2)(x+3)=18yy+1)(y+2)(y+3)仅有4组非平凡整数解(xy)=(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(xy)=(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).
关键词:  不定方程  整数解  递推序列  平方剩余
DOI:10.13718/j.cnki.xdzk.2019.04.013
分类号:O156.2
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).
On the Diophantine Equation 5x(x+1)(x+2)(x+3)=18y(y+1)(y+2)(y+3)
YANG Xiao-liu, MU Quan-wu
School of Science, Xi'an Polytechnic University, Xi'an 710048, China
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).
Key words:  diophantine equation  integer solution  recurrent sequence  quadratic residue
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