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摘要: Let $H=\{1,5,9,\cdots\}$ be the arithmetic progression of all positiveintegers of the form $4k+1$.Elements of $H$ are called Hilbertnumbers.Show that ... 阅读全文
posted @ 2012-12-01 03:24 叶卢庆 阅读(183) 评论(0) 推荐(0)
摘要: Let $H=\{1,5,9,\cdots\}$ be the arithmetic progression of all positiveintegers of the form $4k+1$.Elements of $H$ are called Hilbertnumbers.Show that ... 阅读全文
posted @ 2012-12-01 03:24 叶卢庆 阅读(147) 评论(0) 推荐(0)
摘要: For $n\geq 2$,the rational number \begin{equation}\label{eq:343242} 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} \end{equation}is not an integer.... 阅读全文
posted @ 2012-12-01 01:36 叶卢庆 阅读(154) 评论(0) 推荐(0)
摘要: For $n\geq 2$,the rational number \begin{equation}\label{eq:343242} 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} \end{equation}is not an integer.... 阅读全文
posted @ 2012-12-01 01:36 叶卢庆 阅读(153) 评论(0) 推荐(0)
摘要: Let $a\mathbf{Z}=\{ax:x\in\mathbf{Z}\}$ denote the set of allmultiples of $a$.Prove that for any integers $a_1,\cdots,a_k$,\begin{equation} \bigcap_{... 阅读全文
posted @ 2012-11-30 03:04 叶卢庆 阅读(146) 评论(0) 推荐(0)
摘要: Let $a\mathbf{Z}=\{ax:x\in\mathbf{Z}\}$ denote the set of allmultiples of $a$.Prove that for any integers $a_1,\cdots,a_k$,\begin{equation} \bigcap_{... 阅读全文
posted @ 2012-11-30 03:04 叶卢庆 阅读(113) 评论(0) 推荐(0)
摘要: Let $a$ and $b$ be positive integers such that\begin{equation} a+b=57\end{equation}and\begin{equation} [a,b]=680\end{equation}Find $a$ and $b$.Solve... 阅读全文
posted @ 2012-11-30 02:24 叶卢庆 阅读(128) 评论(0) 推荐(0)
摘要: Let $a$ and $b$ be positive integers such that\begin{equation} a+b=57\end{equation}and\begin{equation} [a,b]=680\end{equation}Find $a$ and $b$.Solve... 阅读全文
posted @ 2012-11-30 02:24 叶卢庆 阅读(128) 评论(0) 推荐(0)
摘要: Prove that for any positive integers $a$ and $b$,\begin{equation} [a,b]=\frac{ab}{(a,b)}\end{equation}该命题根据容斥原理很容易得证.两个数的最小公倍数相当于两个集合的并,两个数的最大公约数相当于两... 阅读全文
posted @ 2012-11-30 01:19 叶卢庆 阅读(115) 评论(0) 推荐(0)
摘要: Prove that for any positive integers $a$ and $b$,\begin{equation} [a,b]=\frac{ab}{(a,b)}\end{equation}该命题根据容斥原理很容易得证.两个数的最小公倍数相当于两个集合的并,两个数的最大公约数相当于两... 阅读全文
posted @ 2012-11-30 01:19 叶卢庆 阅读(112) 评论(0) 推荐(0)
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