问题及解答

观察

\[
\frac{153}{1001}=\frac{1}{8}+\frac{1}{36}+\frac{1}{14415}+\frac{1}{346305960},
\]

\[
\frac{153}{1001}=\frac{1}{7}+\frac{1}{101}+\frac{1}{11234}+\frac{1}{1135768634}+\frac{1}{227153727}+\frac{1}{257994078222798918}.
\]

因此 $\frac{153}{1001}$ 在计算机中也可以表示为 $(8,36,14415,346305960)$. 当然这并不比表示成 $153/1001$ 强多少.

首先不妨验证一下, 其次思考如何用计算机列出所有的表示方法, 即表示为 $\frac{1}{a_1}+\cdots+\frac{1}{a_n}$ 的形式.

这里涉及到的是古埃及分数. 详细可参考 Richard K. Guy 著 Unsolved Problems in Number Theory. 《数论中未解决的问题》. D11.

Remark (on 2021-06-20)

上面两个等式参见 [1] P.183.   第二个等式似乎并不正确.

References:

[1] Raymond Séroul, Programming for Mathematicians, Translated from the French by Donal O'Shea. Springer.

验证一下

\[
\begin{split}
&\frac{1}{8}+\frac{1}{36}+\frac{1}{14415}+\frac{1}{346305960}\\
=&\frac{9+2}{72}+\frac{24024+1}{14415\times 24024}\\
=&\frac{11}{72}+\frac{24025}{14415\times 24024}\\
=&\frac{11}{72}+\frac{4805}{2883\times 24024}\\
=&\frac{11}{72}+\frac{961\times 5}{961\times 3\times 24024}\\
=&\frac{11}{72}+\frac{5}{72072}\\
=&\frac{11\times 1001+5}{72\times 1001}\\
=&\frac{11016}{72\times 1001}\\
=&\frac{153\times 72}{72\times 1001}\\
=&\frac{153}{1001}.
\end{split}
\]

现在有了 Calculator, 我们可以用其分数运算来验证.

>> :mode=fraction
Switch into fraction calculating mode.
e.g., 1/2+1/3 will return 5/6

>> 1/8+1/36+1/14415+1/346305960
in> 1/8+1/36+1/14415+1/346305960

out> 153|1001

------------------------

第二个似乎有问题

>> 1/7+1/101+1/11234+1/1135768634+1/227153727+1/257994078222798918
in> 1/7+1/101+1/11234+1/1135768634+1/227153727+1/257994078222798918

out> 173599003|1135768634

手算验证

\[
\begin{split}
&\frac{1}{7}+\frac{1}{101}+\frac{1}{11234}+\frac{1}{1135768634}+\frac{1}{227153727}+\frac{1}{257994078222798918}\\
=&(\frac{1}{7}+\frac{1}{101})+(\frac{1}{11234}+\frac{1}{1135768634})+(\frac{1}{227153727}+\frac{1}{257994078222798918})\\
=&\frac{101+7}{7\times 101}+\frac{1135768634+11234}{11234\times 1135768634}+\frac{257994078222798918+227153727}{227153727\times 257994078222798918}\\
=&\frac{108}{707}+\frac{1135779868}{12759224834356}+\frac{257994078449952645}{58604316412238310595267386}\\
=&\frac{108}{707}+\frac{50551}{567884317}+\frac{5}{1135768634}\\
=&\frac{108*567884317+707*50551}{707\times 567884317}+\frac{5}{1135768634}\\
=&\frac{61367245793}{401494212119}+\frac{5}{1135768634}\\
=&\frac{86799499}{567884317}+\frac{5}{1135768634}\\
=&\frac{86799499*1135768634+5*567884317}{567884317*1135768634}\\
=&\frac{98584151250535951}{644985194989112978}\\
=&\frac{173599003}{1135768634}
\end{split}
\]