数论 / number theory 中不经意间发现的好玩的

完全数 / perfect numbers

他们的真因子（除了本身以外的所有因子）之和为自己：$6=3+2+1$, $28=14+7+4+2+1$, $496=248+124+62+31+16+8+4+2+1$, $8128=4064+2032+1016+508+254+127+64+32+16+8+4+2+1$, …

质数

19, 199, 1009, 1999 are prime numbers!!!

twin primes / 孪生素数: $3,5,7$，$29,31$，$41,43$，$101,103$，$2027,2029$，$9999971,9999973$

常用的分解：$7×11×13=1001$, $17×59=1003$, $19×53=1007$, $73×137=10001$, $3×7×13×37=10101$, $41×271=11111$

平方数的差：

\begin{aligned}
&1^2=2\times1^2-1,3^2=2\times2^2+1,7^2=2\times5^2-1,17^2=2\times12^2+1,\cdots\\
&\frac{2^2}{2}=1^2+1,\frac{4^2}{2}=3^2-1,\frac{10^2}{2}=7^2+1,\frac{24^2}{2}=17^2-1,\cdots
\end{aligned}

另一些比较特殊的关系

$2\times 84 \times 85=119 \times 120 \Rightarrow 2 \times(1 + 2 + \cdots + 84) = 1 + 2 + \cdots + 119$

$65=8^2+1^2=7^2+4^2=(2^2+1^2)(3^2+2^2),1729=12^3+1^3=10^3+9^3$

$3^2+4^2=5^2,10^2+11^2+12^2=13^2+14^2=365$

$1^2+\cdots+24^2=\frac{24\times25\times49}{6}=4\times25\times49=2^2\times5^2\times7^2=(2\times5\times7)^2=70^2$

需要枚举的

1：$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6} =\frac{1}{2}+\frac{1}{4}+\frac{1}{4} =\frac{1}{3}+\frac{1}{3}+\frac{1}{3}$

½：

\begin{array}{cccc}
\frac{1}{2}&=\frac{1}{3}+\frac{1}{7}+\frac{1}{42}
&=\frac{1}{3}+\frac{1}{8}+\frac{1}{24}
&=\frac{1}{3}+\frac{1}{9}+\frac{1}{18}
&=\frac{1}{3}+\frac{1}{10}+\frac{1}{15}
&=\frac{1}{3}+\frac{1}{12}+\frac{1}{12}\\
&=\frac{1}{4}+\frac{1}{5}+\frac{1}{20}
&=\frac{1}{4}+\frac{1}{6}+\frac{1}{12}
&=\frac{1}{4}+\frac{1}{8}+\frac{1}{8}
&=\frac{1}{5}+\frac{1}{5}+\frac{1}{10}
&=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}
\end{array}

⅓：

\begin{array}{llll}
\frac{1}{3}
&=\frac{1}{4}+\frac{1}{13}+\frac{1}{156}&(1)&&=\frac{1}{4}+\frac{1}{14}+\frac{1}{84}&(2)&&=\frac{1}{4}+\frac{1}{15}+\frac{1}{60}&(3)\\
&=\frac{1}{4}+\frac{1}{16}+\frac{1}{48}&(4)&&=\frac{1}{4}+\frac{1}{18}+\frac{1}{36}&(5)&&=\frac{1}{4}+\frac{1}{20}+\frac{1}{30}&(6)\\
&=\frac{1}{4}+\frac{1}{21}+\frac{1}{28}&(7)&&=\frac{1}{4}+\frac{1}{24}+\frac{1}{24}&(8)&&=\frac{1}{5}+\frac{1}{8}+\frac{1}{120}&(9)\\
&=\frac{1}{5}+\frac{1}{9}+\frac{1}{45}&(10)&&=\frac{1}{5}+\frac{1}{10}+\frac{1}{30}&(11)&&=\frac{1}{5}+\frac{1}{12}+\frac{1}{20}&(12)\\
&=\frac{1}{5}+\frac{1}{15}+\frac{1}{15}&(13)&&=\frac{1}{6}+\frac{1}{7}+\frac{1}{42}&(14)&&=\frac{1}{6}+\frac{1}{8}+\frac{1}{24}&(15)\\
&=\frac{1}{6}+\frac{1}{9}+\frac{1}{18}&(16)&&=\frac{1}{6}+\frac{1}{10}+\frac{1}{15}&(17)&&=\frac{1}{6}+\frac{1}{12}+\frac{1}{12}&(18)\\
&=\frac{1}{7}+\frac{1}{7}+\frac{1}{21}&(19)&&=\frac{1}{8}+\frac{1}{8}+\frac{1}{12}&(20)&&=\frac{1}{9}+\frac{1}{9}+\frac{1}{9}&(21)
\end{array}

尽量记的常数

\begin{aligned}
e=\lim_{x\to\infty}(1+\frac{1}{x})^x=\lim_{y\to0}(1+y)^\frac{1}{y}&=2.718281828459...\\
\pi&=3.14159265359...\\
\sqrt{2}&=1.414213562373095...\\
\sqrt{3}&=1.732050807568877...\\
\sqrt{5}&=2.23606797749979...\\
\sqrt{10}&=3.1622776601683793319988935444327...\\
1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}&≈\frac{\pi^2}{6}≈1.644934066848226436472415166646...
\end{aligned}