Recently, Kai-Liang brought to my attention an extension form of the classic inequality of the arithmetic and geometric means. It is easy to see C2=1. It is not easy to calculate C3. I spent a couple of days on this problem and found the solution. My method is very sta ...
素数基本定理 是描述素数分布的一个非常重要的定理。它表明素数的密度大约是1/log(x),即 不大于x内的素数个数大约为x/log(x) 。 Jacques Solomon Hadamard和Charles-Jean de la Vallée Poussin于1896年按照B. Riemann的思想首次证明。 简单地说,素数基本定理等价于 黎曼zeta函数zeta(s)在实部为1的 ...
I knew this problem in a WeChat group chat. It is said to be a problem for primary-school students. Here is the problem. How many integers with all digits being just 1,2 or 3 and the sum of all digits being 10? For example, 12331 and 22222. We could ask a more general qu ...
Fang and I found enhanced inequalities about the arithmetic mean and the geomertic mean of n positive numbers (see this arXiv entry for details). Ofcourse it is impossible to improve those classic inequalities without new information. Let me explain our ideas. Suppose we k ...