| 关于丢番图方程Dx2+1=can |
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| 引用本文： | 袁平之.关于丢番图方程Dx2+1=can[J].黑龙江大学自然科学学报,2005,22(2):195-197. |
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| 作者姓名： | 袁平之 |
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| 作者单位： | 中山大学数学与计算科学学院,广东,广州,510275 |
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| 基金项目： | Supported by Guangdong Provincial Natural Science Foundation(04009801) |
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| 摘 要： | 设c和a为正整数,D为与ca互素的正整数.记N(D;c,a)为方程Dx2+1=can的解(x,n)的个数,其中x及n是正整数.利用Nagell和Ljunggren的一个结果和Wallker的一个结果,证明了除N(2;1,3)=3,N(6;1,7)=N(7;1,2)=2和N(D;1,b2-1)=2,其中b＞1为正整数且Ds2=b2-2,s为整数,均有N(D;1,a)≤1;除N(2;1,3)=3,均有N(D;c,a)≤2.
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| 关 键 词： | 指数丢番图方程 二次方程 阶 |
On the Diophantine equation Dx2 + 1 = can |
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| YUAN Ping-zhi.On the Diophantine equation Dx2 + 1 = can[J].Journal of Natural Science of Heilongjiang University,2005,22(2):195-197. |
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| Authors: | YUAN Ping-zhi |
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| Abstract: | Let c and a be positive integers, and let D be a positive integer coprime with ca. Denote by N(D;c,a) the number of solutions (x,n) of the equation Dx2 + 1 = can in positive integers x and n.By using a result of Nagell and Ljunggren and a result of Walker, it is shown that: N( D;1, a) ≤ 1 except for N(2; 1,3 ) = 3,N(6; 1,7 ) = N(7; 1,2) = 2, and N(D; 1, b2 - 1 ) = 2, where b ＞ 1 is a positive integer and Ds2 = b2 -2 for some integer s; N(D;c,a) ≤2 except for N(2;1,3) =3. |
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| Keywords: | exponential diophantine equations quadratic equations order |
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