### On the Periodicity of Chinese Sexagenary Indices

I have written a short note on the periodicity of Chinese sexagenary indices (traditionally known as bazi 八字) after reading a book by Hee on the same subject, many years ago. I was motivated primarily by a section of her book. In page 22, Hee wrote:

...because of the numerous possible combinations, it takes 60 years for the same set of year pillars to repeat itself (by comparison, a set of month pillars repeats itself after just five years). Therefore, if you have a certain day and time, the set of four pillars will repeat itself in 60 years. However, since the same day may not appear in exactly the same month - and even if it is in the same month, the day may not be found in the same half month - it takes 240 years before the identical four pillars appear again.
Basically, Hee's claims are as follows:

1. When considered independently, the periodicity of the year-index is 60 years, which is obviously correct.
2. When considered independently, the periodicity of the month-index is 5 years, which is also correct.
3. In general, the complete set of sexagenary indices will repeat itself after 240 years. This claim is incorrect and will be illustrated by a counterexample.
It appeared in the paragraph quoted above that Hee believes that the interval between two successive sexagenary indices is regularly maintained at 240 years. I believe this assertion is flawed when I first read it and I believe I know enough math to check the validity of this claim. First of all, it is easy to formulate an expression for the interval $$g$$ between two successive sexagenary indices. It is irregular and it is given by
$$g = 60(365\ell_1 + \ell_2)$$
As a numerical example, the sexagenary indices (year, month, and day) associated with March 18, 1984 are given by (甲子, 丁卯, 辛亥). Using calendrical algorithm by Reingold and Dershowitz, we can easily compute other dates with similar sexagenary indices, and they are tabulated in the table below.

In the table, we see that the indices are repeated 60 years after 1984, namely on March 3, 2044. However, after 2044, the next identical indices appear only after 360 years in April 5, 2404. This clearly contradicts Hee's assertion (by the way, Hee is an accountant by training. It is not clear to me as to who is the originator of this theory, could this idea came from her mentor: Raymond Lo of Hong Kong?). Another interesting observation is that some entries in the table share identical month and day with March 18, 1984. (i.e. March 18, 4 and March 18, 3964). They appear approximately 1980 years (or exactly 723,180 days) before and after March 18, 1984. This is no coincidence because suppose $$f$$ and $$f + g'$$ are two successive fixed day numbers with identical month and day, then it can be shown that the interval $$g'$$ is given by
$$g' = 365\ell_3 + 366\ell_4$$
For $$g$$ and $$g'$$ to coincide, we need to solve the following Diophantine equation
$$60(365\ell_1 + \ell_2) = 365\ell_3 + 366\ell_4$$
of which one of the solution is $$(\ell_1, \ell_2, \ell_3, \ell_4) = (33, 8, 1575, 402)$$. The following additional constraints apply when solving the Diophantine equation: The ratio of $$\ell_4$$ to $$\ell_3$$ is approximately $$\frac{1}{4}$$, and that for large $$\ell_1$$, we have $$\ell_1 = \lfloor \frac{1}{4}\ell_2 \rfloor$$.