Abstract
The sexagenary cycle names years by pairing one of ten heavenly stems (jia, yi, bing and so on) with one of twelve earthly branches (zi, chou, yin and so on). Advancing both by one each year, how many pairs occur is counted. No new theorem or law is claimed.
First, the pairs that occur are $60$, not $120$——collecting pairs over $120$ years, $60$ occurred (share $0.500000$); the cycle repeats after $60$ years and no new pair appears afterwards. Jia--zi comes again only after $60$ years, the return known as kanreki in Japan (Section 2).
Second, only pairs of matching parity occur——numbering the stems and branches from $0$, all $60$ pairs that occurred have indices of the same parity, and there are exactly $60$ such pairs. The pair of jia ($0$) and chou ($1$) never comes (Section 2).
Third, and this is the core. The separator is the greatest common divisor of the two periods——each year both indices advance by one, so the difference of the indices keeps its remainder on division by $2$, the greatest common divisor of $10$ and $12$. With greatest common divisor $1$ every pair occurs; with $g$, only $1/g$ of them (Section 3).
Fourth, in the Maya calendar only a fifth occur——the pairs of $13$ numbers and $20$ day names (the tzolk’in) have greatest common divisor $1$, and all $260$ occur. The pairs of that $260$-day count and the $365$-day count have greatest common divisor $5$; only $18980$ of $94900$ occur (share $0.200000$), repeating after $18980$ days ($52$ years of $365$ days) (Section 3).
Fifth, three pairwise coprime periods make everything occur——the three periods of $28$, $19$ and $15$ years (the Julian period) have greatest common divisor $1$ in every pair, and in $7980$ years all $7980$ triples occur once each (Section 3).
Sixth, two routes agree——the number of pairs collected by running through the years and the number counted only by the condition “the difference of the indices leaves $0$ on division by the greatest common divisor” agreed for the sexagenary cycle and for the Maya calendar (Section 3).
Keywords: sexagenary cycle, Chinese remainder theorem, greatest common divisor, calendar, Calendar Round.
Scope (what is not claimed)
No new theorem or law is claimed——the theorem fixing a number from two remainders (the Chinese remainder theorem), that the sexagenary cycle repeats after $60$ years, that the Maya calendar round is $52$ years, and the Julian period (Scaliger, $1583$) are all known. The history of calendars is not treated——reforms, leap days, regional differences, the origin of the stems and branches, and how the Maya count is aligned to present dates are not the subject. Conventions of year numbering are not discussed——all indices start from $0$, and only how periods combine is counted. Relation to earlier papers: Paper 319 showed that a calendar is three stacked approximations fitting human numbers to natural periods——this paper is not about approximation; it counts how many pairs occur when two or three periods fixed by people are combined. What is added is setting the sexagenary cycle, the tzolk’in, the calendar round and the Julian period side by side under one count, matching the route of running through the years with the route of counting by the condition, and placing the separator on the greatest common divisor of the periods and checking in numbers that the share occurring is $1/g$.
1. Notation and the Question
Setting: number the stems jia, yi, bing, ding, wu, ji, geng, xin, ren, gui from $0$ to $9$, and the branches zi, chou, yin, mao, chen, si, wu, wei, shen, you, xu, hai from $0$ to $11$. The pair of year $y$ (with $y=0$ the year jia--zi) is $(a,b)$, the remainders of $y$ on division by $10$ and by $12$.
In general: when two counts with periods $m_1,m_2$ advance by one together, the pairs $(y \bmod m_1, y \bmod m_2)$ that occur are counted.
Question: of the $m_1m_2$ possible pairs, how many occur, and what decides it?
2. Counting by Running Through the Years
Computation 2.1: pairs were collected by running through the years (up to three cycles, confirming that no new pair appears after one cycle).
| periods combined | possible pairs | pairs occurring | share | cycle |
|---|---|---|---|---|
| stems and branches ($10$ and $12$) | $120$ | $60$ | $0.500000$ | $60$ years |
| tzolk’in ($13$ and $20$) | $260$ | $260$ | $1.000000$ | $260$ days |
| calendar round ($260$ and $365$ days) | $94900$ | $18980$ | $0.200000$ | $18980$ days |
| Julian period ($28$, $19$, $15$ years) | $7980$ | $7980$ | $1.000000$ | $7980$ years |
Observation 2.2 the sexagenary cycle has only $60$ pairs. Jia--zi comes again after $60$ years, and in between each of the $60$ pairs occurs once——half of the $120$ candidates never come, however many years pass.
Observation 2.3 the $60$ pairs that occur are those whose indices have the same parity. Jia--zi $(0,0)$, yi--chou $(1,1)$, bing--yin $(2,2)$ follow in turn, and after gui--you $(9,9)$ comes jia--xu $(0,10)$. Pairs of matching parity number $5\times6+5\times6=60$ and coincided exactly with those that occurred——jia--chou $(0,1)$ never comes, since its parities do not match.
3. The Separator
Observation 3.1 this is the core: the separator is the greatest common divisor of the two periods. $a$ is the remainder of $y$ on division by $10$, so it has the same parity as $y$ ($10$ being divisible by $2$); likewise $b$ has the same parity as $y$ ($12$ being divisible by $2$)——so $a$ and $b$ always have the same parity. $2$ is the greatest common divisor of $10$ and $12$, and in general $a$, $b$ and $y$ all leave the same remainder on division by the greatest common divisor $g$.
Computation 3.2: independently of running through the years, pairs were counted only by the condition “$a-b$ leaves $0$ on division by the greatest common divisor $g$”.
| periods | greatest common divisor $g$ | pairs meeting the condition | least common multiple |
|---|---|---|---|
| $10$ and $12$ | $2$ | $60$ | $60$ |
| $13$ and $20$ | $1$ | $260$ | $260$ |
| $260$ and $365$ | $5$ | $18980$ | $18980$ |
The numbers counted by the condition agreed with those collected by running through the years, for the sexagenary cycle and for the calendar round. Each equals the least common multiple $m_1m_2/g$, a share $1/g$ of the candidates.
Observation 3.3 every pair meeting the condition occurs. For a pair $(a,b)$ of matching parity, there is exactly one year among $60$ that leaves $a$ on division by $10$ and $b$ on division by $12$——this is the theorem fixing a number from two remainders (the Chinese remainder theorem). When the greatest common divisor is $1$, the condition disappears and every pair occurs.
Observation 3.4 the Maya calendar is the same mechanism in two stages. $13$ and $20$ have greatest common divisor $1$, so all $260$ pairs line up in the $260$-day tzolk’in. That tzolk’in and the $365$-day count have greatest common divisor $5$, so only a fifth, $18980$ pairs, occur, repeating after $18980$ days, that is $52$ years of $365$ days——in the first stage nothing vanishes; in the second, four fifths vanish. The only difference is the greatest common divisor.
Observation 3.5 the Julian period is built so that nothing vanishes. $28$, $19$ and $15$ have greatest common divisor $1$ in every pair, and in $7980$ years all $7980$ triples occur once each——Scaliger built it as the length within which the triple of indices fixes a single year.
Observation 3.6 beside Paper 319. Paper 319 found that the Julian calendar drifts by one day in $128$ years and the Gregorian by one day in $3226$ years, and showed a calendar to be three stacked approximations——there the issue was approximating natural periods with human numbers. Here all periods are integers fixed by people, with no approximation, and whether pairs vanish is decided only by the greatest common divisor of those integers.
Observation 3.7 what is not said. The history of calendars, reforms, leap days, the origin of the stems and branches, and how the Maya count is aligned to present dates are not treated——what is shown is that the number of pairs occurring when periods are combined is decided by the greatest common divisor, checked by two routes.
4. Roll Call and Closing
- $60$ of the $120$ possible pairs occur (Computation 2.1): share $0.500000$.
- the cycle repeats after $60$ years with no new pair (Computation 2.1): kanreki.
- only pairs of matching parity occur (Observation 2.3): jia--chou never comes.
- the separator is the greatest common divisor of the periods (Observation 3.1): the core.
- pairs counted by the condition match those from the years (Computation 3.2): two routes.
- the share occurring is $1/g$ (Computation 3.2): as many as the least common multiple.
- all $260$ pairs of the tzolk’in occur (Computation 2.1): greatest common divisor $1$.
- the calendar round has $18980$ of $94900$ (Computation 2.1): greatest common divisor $5$, $52$ years.
- all $7980$ triples of the Julian period occur (Computation 2.1): built so nothing vanishes.
- decided by the greatest common divisor of integers, not by approximation (Observation 3.6): the difference from Paper 319.
Closing: The sexagenary cycle names years by advancing the ten stems and the twelve branches by one each year. There are $120$ possible pairs, but only $60$ occur——running through the years, the cycle repeats after $60$ years and no new pair appears. Only pairs whose indices have the same parity occur, and jia--chou never comes. The separator is the greatest common divisor of the two periods——since both indices advance together, the difference of the indices keeps its remainder on division by the greatest common divisor. With greatest common divisor $g$, the pairs occurring are $1/g$ of the candidates, and the count from the years agreed with the count by the condition. The same mechanism is in the Maya calendar——$13$ and $20$ have greatest common divisor $1$ and all $260$ pairs occur, while $260$ and $365$ days have $5$ and only $18980$ of $94900$ occur. The Julian period, built from $28$, $19$ and $15$ years, has greatest common divisor $1$ in every pair, and all $7980$ triples occur. Placed among the earlier papers——unlike the approximations of the calendar treated in Paper 319, all periods here are integers, and whether pairs vanish is decided by the greatest common divisor alone. To be honest——this is a count under a known theorem, and neither the history of calendars nor the alignment of the Maya count is touched.
References
- Aveni, A. F. (2001). Skywatchers: A Revised and Updated Version of Skywatchers of Ancient Mexico. University of Texas Press.
- Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
- Richards, E. G. (1998). Mapping Time: The Calendar and its History. Oxford University Press.
- Scaliger, J. J. (1583). Opus novum de emendatione temporum.
- Yamagishi, Y. (2026). A Calendar Is Three Approximations Stacked [Paper 319]. Zenodo. 10.5281/zenodo.22208933
Reference
Yamagishi, Y. (2026). The Sexagenary Cycle Has Sixty Pairs, Not a Hundred and Twenty [Paper 605]. YOI: https://www.yamagishiyuuki.com/yoi/605
DOI: 10.5281/zenodo.23014440
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 605. This paper is a preprint and has not been peer reviewed.