蔡小群.关于不定方程x2+4n=y11的整数解[J].重庆工商大学学报(自然科学版),2021,38(1):99-104
CAI Xiao-qun.The Integer Solution of the Diophantine Equation x2+4n=y11[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2021,38(1):99-104
关于不定方程x2+4n=y11的整数解
The Integer Solution of the Diophantine Equation x2+4n=y11
  
DOI:
中文关键词:  不定方程  整数解  代数数论
英文关键词:Diophantine equation  integer solution  algebraic number theory
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作者单位
蔡小群 华南师范大学 数学科学学院广州 510631 
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中文摘要:
      应用代数数论以及同余法等初等方法讨论不定方程x2+4n=y11的整数解情况,证明了不定方程x2+4n=y11在x为奇数,n≥1时无整数解; 不定方程x2+4n=y11在n∈{1,8,9,10}时均无整数解; 不定方程x2+4n=y11有整数解的充要条件是n≡0(mod 11)或n≡5(mod 11),且当n≡0(mod 11)时,其整数解为(x,y)=(0,4m);当n≡5(mod 11)时,其整数解为 (x,y)=(±211m+5,22m+1), 这里的m为非负整数, 验证了k=11时猜想1成立。
英文摘要:
      The Diophantine equation,as the oldest branch of nmber theory,is very rich in content.The so-called Diophantine equation refers to the equation where the number of unknowns is more than the number of equations.The integer solution of x2+4n=y11 is studied by algebraic number theory and congruence and so on.It proves that the Diophantine equation x2+4n=y11 has no integer solution when x≡1(mod 2),n≥1, and shows that the Diophantine equation x2+4n=y11(n∈{1,8,9,10})has no integer solution.So the Diophantine equation x2+4n=y11 has integer solutions if and only if n≡0,5(mod11),and (x,y)=(0,4m) when n=11m, (x,y)=(211m+5,22m+1) when n=11m+5, where m is a non negative integer.It implies that Conjecture 1 holds for k=11.
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