| 摘要: |
| 应用代数数论以及同余法等初等方法讨论不定方程x2+4n=y11的整数解情况，证明了不定方程x2+4n=y11在x为奇数，n≥1时无整数解； 不定方程x2+4n=y11在n∈{1,8,9,10}时均无整数解; 不定方程x2+4n=y11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)=(0,4m)；当n≡5(mod 11)时,其整数解为 (x,y)=(±211m+5,22m+1)， 这里的m为非负整数， 验证了k=11时猜想1成立。 |
| 关键词: 不定方程 整数解 代数数论 |
| DOI： |
| 分类号: |
| 基金项目: |
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| The Integer Solution of the Diophantine Equation x2+4n=y11 |
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CAI Xiao-qun
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School of Mathematics, South China Normal University, Guangzhou 510631,China
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| Abstract: |
| 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 x2+4n=y11 is studied by algebraic number theory and congruence and so on.It proves that the Diophantine equation x2+4n=y11 has no integer solution when x≡1(mod 2),n≥1, and shows that the Diophantine equation x2+4n=y11(n∈{1,8,9,10})has no integer solution.So the Diophantine equation x2+4n=y11 has integer solutions if and only if n≡0,5(mod11),and (x,y)=(0,4m) when n=11m, (x,y)=(211m+5,22m+1) when n=11m+5, where m is a non negative integer.It implies that Conjecture 1 holds for k=11. |
| Key words: Diophantine equation integer solution algebraic number theory |
