ABA Numbers

The number (27783) associated with my diurnal age today has the property that it can be expressed as:$$27783=3\times {21}^3$$Numbers like this are called ABA numbers because they can be expressed in the form:$${\text{AB}}^{\text{A}}\text{ for A, B > 1}$$Up to 40,000, the ABA numbers are as follows (link):

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592, 2738, 2888, 3000, 3042, 3200, 3362, 3528, 3698, 3872, 3993, 4050, 4232, 4374, 4418, 4608, 4802, 5000, 5120, 5184, 5202, 5408, 5618, 5832, 6050, 6272, 6498, 6591, 6728, 6962, 7200, 7442, 7688, 7938, 8192, 8232, 8450, 8712, 8978, 9248, 9522, 9604, 9800, 10082, 10125, 10240, 10368, 10658, 10952, 11250, 11552, 11858, 12168, 12288, 12482, 12800, 13122, 13448, 13778, 14112, 14450, 14739, 14792, 15138, 15309, 15488, 15625, 15842, 16200, 16384, 16562, 16928, 17298, 17496, 17672, 18050, 18432, 18818, 19208, 19602, 20000, 20402, 20577, 20808, 21218, 21632, 22050, 22472, 22528, 22898, 23328, 23762, 24000, 24200, 24576, 24642, 25088, 25538, 25992, 26244, 26450, 26912, 27378, 27783, 27848, 28322, 28800, 29282, 29768, 30258, 30752, 31250, 31752, 31944, 32258, 32768, 33282, 33800, 34322, 34848, 35378, 35912, 36450, 36501, 36992, 37538, 38088, 38642, 38880, 39200, 39762, 40000

648 is the first ABA number with two representations:$$648=2\times {18}^2=3\times 6^3$$The smallest number with three such representations is 344373768:$$344373768=8\times 9^8=3\times {486}^3=2\times {13122}^2$$The smallest Pythagorean triples made of ABA numbers are:$$(98304,131072,163840)\text{ and }(229376,786432,819200)$$which correspond to:$$(3\times {32}^3,2\times {256}^2,5\times 8^5)\text{ and }(14\times 2^{14},3\times {64}^3,2\times {640}^2)$$

Some Special Sphenic Numbers

Today I turned 27782 days old and one of the properties of 27782 is that it is sphenic since:$$27782=2\times 29\times 479$$However, looking at the digital roots of the number and its factor we notice an interesting fact:$$\underset{8}{\underbrace{27782}}=\underset{2}{\underbrace{2}}\times \underset{2}{\underbrace{29}}\times \underset{2}{\underbrace{479}}$$The respective digital roots are shown under the number and its factors and we see that the digital roots of the factors (2) multiply together to give the digital root of the number (8). The only other way this can occur is if all the digital roots are 1.

In the range up to 100,000 there are 230 sphenic numbers with the property that the digital roots of the factors multiply to give the digital root of the number. These numbers are (permalink):

638, 1034, 1826, 2222, 2726, 3014, 3806, 4202, 4814, 4994, 5786, 5858, 6182, 6974, 7766, 7802, 7946, 8558, 9494, 9746, 10034, 10142, 10538, 11078, 12518, 12878, 12914, 13166, 14102, 14498, 14894, 14993, 15254, 16262, 16298, 16766, 17954, 18062, 18386, 18458, 18854, 20042, 20438, 20474, 20834, 21338, 21626, 22418, 22562, 22742, 24002, 24398, 24722, 25586, 25694, 25982, 26414, 26477, 26738, 26774, 27674, 27782, 28358, 28718, 28754, 29798, 29942, 31526, 31706, 31922, 32219, 32714, 33002, 33182, 33506, 34046, 34298, 34946, 35486, 36566, 36674, 37178, 37682, 37862, 38222, 38582, 39266, 39842, 40634, 41642, 41822, 42911, 43334, 43406, 43658, 43703, 44594, 45026, 45386, 45782, 45854, 46178, 46646, 47366, 47402, 47618, 48554, 48662, 49346, 49706, 50534, 51319, 51326, 51722, 52217, 52334, 52622, 52838, 53126, 53306, 53486, 53702, 53882, 54098, 54494, 54926, 55178, 55187, 55682, 56078, 56762, 57014, 57662, 58454, 58598, 59102, 59246, 59642, 60038, 60254, 60929, 61622, 61946, 62018, 62198, 62414, 63278, 63638, 63998, 64034, 64322, 64394, 64574, 65186, 65978, 66086, 67454, 67958, 68498, 70586, 70829, 71306, 71522, 72062, 72413, 73106, 73538, 73898, 74762, 75086, 75806, 75878, 76274, 76526, 76627, 76994, 77174, 77858, 78254, 78542, 78578, 78866, 78938, 79046, 79514, 80558, 81422, 81818, 83114, 83897, 84158, 84986, 85382, 85634, 86174, 86246, 86714, 86858, 87326, 87758, 88154, 88334, 89018, 89281, 89441, 89486, 89639, 89738, 90422, 90926, 90998, 92213, 92402, 93122, 93302, 93554, 93698, 94454, 95678, 95786, 96686, 96722, 96758, 97226, 97262, 97442, 98054, 98747, 98846, 99818

However, of these only three have digital roots that are all equal to 1. These are:$$\begin{array}{rl}51319& =19\times 37\times 73\\ 76627& =19\times 37\times 109\\ 89281& =19\times 37\times 127\end{array}$$As can be seen, two of the factors (19 and 37) are the same for all three numbers. If we extend the range to one million, these two factor and 73 (the reversal of 37) make frequent appearances.

If we relax the requirement that the digital roots of the factors must be equal and require only the the digital roots of the factors multiply together to give the digital root of the number, then we find that 1969 numbers satisfy and that will include the 230 numbers mentioned earlier (permalink). An example would be 27813:$$\underset{3}{\underbrace{27813}}=\underset{3}{\underbrace{3}}\times \underset{1}{\underbrace{73}}\times \underset{1}{\underbrace{127}}$$

The Nautical Mile

Today I turned 27780 days old and one of the properties of this number is that it is a multiple of 1852:$$27780=15\times 1852$$The obvious question is so what? Well, 1852 turns out to be the length of a nautical mile in metres and numbers that are multiples of it belong to a special sequence, OEIS A303272. The initial members of the sequence are:

0, 1852, 3704, 5556, 7408, 9260, 11112, 12964, 14816, 16668, 18520, 20372, 22224, 24076, 25928, 27780, 29632, 31484, 33336, 35188, 37040, 38892, 40744, 42596, 44448, 46300, 48152, 50004, 51856, 53708, 55560, 57412, 59264, 61116, 62968, 64820, 66672, 68524

Figure 1

So if one day of my life represents a metre then I have today travelled 15 nautical miles on my journey through life. However, let's look more closely at how this figure of 1852 comes about. See Figure 1 and here are some excerpts from Wikipedia:

By the mid-19th century, France had defined a nautical mile via the original 1791 definition of the metre, one ten-millionth of a quarter meridian. So:$$\frac{10,000,000}{90\times 60}\text{m }=1,851.85\text{ m }\approx 1,852\text{ m }$$became the metric length for a nautical mile. France made it legal for the French Navy in 1906, and many metric countries voted to sanction it for international use at the 1929 International Hydrographic Conference. In 1929 the international nautical mile was defined by the First International Extraordinary Hydrographic Conference in Monaco as exactly 1,852 metres (which is 6,076.12 ft). The United States did not adopt the international nautical mile until 1954. Britain adopted it in 1970. The derived unit of speed is the knot, one nautical mile per hour.

OEIS A303272 suggests a second similar sequence based on 1760, the number of yards in a mile. This would produce the follow initial members:

0, 1760, 3520, 5280, 7040, 8800, 10560, 12320, 14080, 15840, 17600, 19360, 21120, 22880, 24640, 26400, 28160, 29920, 31680, 33440, 35200, 36960, 38720, 40480, 42240, 44000, 45760, 47520, 49280, 51040, 52800, 54560, 56320, 58080, 59840, 61600, 63360, 65120, 66880, 68640, 70400

However, this sequence is not listed in the OEIS. Multiples of 5280, the number of feet in a mile, are included of course in the above sequence of numbers.

Descent to Zero

The number associated with my diurnal age today, 27779, has the interesting property that it is the smallest number that takes 22 steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". The possible concatenatable pairs and their products for 27779 are:

• 2 * 7779 = 15558
• 27 * 779 = 21033
• 277 * 79 = 21883
• 2777 * 9 = 24993

We see that 24993 is the maximum product and we repeat the process using this number as our starting point:

• 2 * 4993 = 9986
• 24 * 993 = 23832
• 249 * 93 = 23157
• 2499 * 3 = 7497

The maximum product is 23832 and so this becomes the new number and the process continues until we reach 0 after 22 steps. The progression is as follows (permalink):

27779, 24993, 23832, 19136, 11478, 9176, 6916, 5496, 5184, 4284, 3528, 2816, 1686, 1376, 988, 792, 644, 264, 128, 96, 54, 20, 0

27779 is a member of OEIS A035932: smallest number that takes $n$ steps to reach 0 under "$k\to$ max product of two numbers whose concatenation is $k$". The initial members of this sequence are:

0, 1, 11, 26, 39, 77, 117, 139, 449, 529, 777, 1117, 2229, 2982, 4267, 4779, 5319, 5919, 8693, 12699, 14119, 17907, 27779, 47877, 80299, 103199, 135199, 274834, 293938, 312794, 606963, 653993, 773989, 1160892, 1296741, 1616696, 1986576

This permalink will check for the smallest number once the number of steps is specified. When writing the code the convention is that you add a condition to handle single-digit numbers. A common rule for sequences like this that aim to reach 0 is that single-digit numbers (other than 0) map to 0 in the next step. So looking at the above sequence we see following progressions to 0 beginning with 0 that requires zero steps (permalink):

• 0
• 1, 0
• 11, 1, 0
• 26, 12, 2, 0
• 39, 27, 14, 4, 0
• 77, 49, 36, 18, 8, 0
• 117, 77, 49, 36, 18, 8, 0
• 139, 117, 77, 49, 36, 18, 8, 0
• 449, 396, 288, 224, 88, 64, 24, 8, 0
• 529, 468, 368, 288, 224, 88, 64, 24, 8, 0
• 777, 539, 477, 329, 288, 224, 88, 64, 24, 8, 0
• 1117, 777, 539, 477, 329, 288, 224, 88, 64, 24, 8, 0
• 2229, 1998, 1862, 1116, 666, 396, 288, 224, 88, 64, 24, 8, 0
• 2982, 2378, 1896, 1728, 1376, 988, 792, 644, 264, 128, 96, 54, 20, 0

More On Sums of Digits

Not long ago, in December of 2024, I created a post titled Prime Sums Of Digits, Digits Squared And Digits Cubed in which I looked at numbers that have a sum of digits, a sum of digits squared and a sum of digits cubed that are all prime. There are 1985 such numbers in the range up to 40000 and 322 of them are prime themselves.

In this post I want to look at numbers that not only meet the just mentioned criteria but have the additional property that, when these sums are added to the original number, the result is also prime. Here are the criteria that such numbers must meet with $\text{SOD}$ standing for Sum Of Digits:

• $\text{SOD}$ of number is prime
• ${\text{SOD}}^2$ of number is prime
• ${\text{SOD}}^3$ of number is prime
• number + $\text{SOD}$ is prime
• number + ${\text{SOD}}^2$ is prime
• number + ${\text{SOD}}^3$ is prime

As it turns out, there are only 19 such numbers in the range up to 40000 and only the first two of them are prime (permalink): 

11, 101, 166, 4490, 4528, 4630, 6016, 8254, 8788, 10066, 12422, 13166, 18284, 18688, 20854, 25570, 31166, 32518, 36064

The first composite number is 166 so let's check that it satisfies the criteria:

• $\text{SOD}$ is prime: 13
• ${\text{SOD}}^2$ is prime: 73
• ${\text{SOD}}^3$ is prime: 433
• 166 + 13 = 179 is prime
• 166 + 73 = 239 is prime
• 166 + 433 = 599 is prime

Interestingly, six of the above numbers survive the addition of a further criterion, namely that:

•  ${\text{SOD}}^4$ is prime
• number + ${\text{SOD}}^4$ is prime

These numbers are 11, 101, 4528, 6016, 10066, 20854 (permalink) and we have to look at much larger numbers to find any that satisfy yet another criterion, namely:

•  ${\text{SOD}}^5$ is prime
• number + ${\text{SOD}}^5$ is prime

The only two numbers that satisfy this additional criteria are 100001 and 104930. The former number is not prime but composite since 100001 = 11 x 9091. It would be interesting to see how much further this process could be taken but we would then be looking at very large numbers indeed.

Quadruple Seven

There has been an uncharacteristic hiatus in my posts during April. My last post was on April 3rd on the occasion of my 76th birthday, and I've been ill ever since. However, today is a very special day and marks the appearance of four sevens in the number associated with my diurnal age. The number is 27777.

This number has some interesting properties. Firstly, it is the last so-called $\mathbf{\text{iban}}$ number till 40000. The definition follows:

A number is called iban if its name (in English) does not contain the letter "i". Assuming that the name of every power of 10 greater than ${10}^5$ ends in "-illion" (like million, billion, trillion, etc.), then the iban numbers are finite. Counting 0 (zero) there are 30276 of them, the largest being 777777. Iban numbers belong to the same family as aban numbers, eban numbers, oban numbers, and uban numbers. See my blog post Iban Numbers.

So whether we write "twenty seven thousand seven hundred seventy seven" (American style) or "twenty seven thousand seven hundred and seventy seven" (British style", the iban property is unaffected.

A second, related property of the number is that it is a member of OEIS A002810 with $n=16$.

 A002810: smallest number containing $n$ syllables in UK English.

The associated sequence in American English is OEIS A045736 where the "and" is omitted and thus the number of syllables required is one less. There is an interesting paradox associated with this OEIS sequence that goes like this (to quote from the OEIS comments) and involves one of its members (1117777):

a(19) = 111777 is precisely the number used for Berry's paradox. In UK English the name of the number 111777 requires 19 syllables -- "one hundred and eleven thousand seven hundred and seventy-seven" -- and it's exactly the smallest number containing 19 syllables in UK English.

The paradox occurs when we consider that this integer is "the least integer not nameable in fewer than nineteen syllables" yet 111777 has just now been defined in eighteen syllables with this last sentence. So there is a contradiction, because the smallest integer expressible in no fewer than nineteen syllables can be expressed in eighteen syllables. This contradiction is Berry's paradox.

It can be noted that 27777, though not prime itself, contains only prime digits (2 and 7). Interestingly we have:$$\sqrt{27777}\approx 166.66\dots$$Thus there is a run of four sixes in the decimal expression of the square root.

While composing this post I ran into a problem with getting the vertical bar to cover the number. This was part of a broader problem with my LaTeX displaying properly on my Macbook Air when using Chrome. For example, mathematical variables would not display in italics but remained upright. These were just two of several other minor glitches and so I got Gemini to examine my HTML code and suggest a solution. It did and the fix worked just fine.

However, another problem soon emerged in that the new line command wasn't working and was being ignored in my updated MathJax code. Again, Gemini was quick to offer a solution.

Using \\ outside a Multi-Line Environment: 

If you're just putting \\ inside $ ... $ or $$...$$ without an environment, MathJax will likely ignore it because those delimiters are meant for single mathematical expressions, even if they span multiple lines of code.

Solution: 

You need to wrap your multi-line math content in an appropriate environment. Common ones supported by MathJax are (see Figure 1):

• align: For aligning equations at a specific point (e.g., the equals sign).
• gather: For centering multiple equations without alignment.
• array: For creating arrays or matrices, where \\ separates rows. 

Figure 1

Again very helpful and just another example of the usefulness of AI. I'm hoping that soon I'll be able to use Gemini to organize, in various ways, the hundreds of posts that I've made in this Mathematics blog over the years. On a humorous note it was instructive to discover my ignorace about Blogger's user interface despite having used it for so many years.

You've hit upon a common point of confusion with website editors like Blogger. The composing window where you write your posts is part of Blogger's administrative interface. The styles (like fonts, sizes, and sometimes even line spacing) you see while typing in that editor are controlled by Blogger's own styling for the editor itself, not by the theme you apply to your public blog.

Think of it like writing a document in a word processor (the Blogger editor) versus how that document looks when you save it as a PDF or print it (your live blog). The word processor might use a default font for typing, but the final output uses the fonts you've specified in the document's formatting.

Therefore, you cannot directly change the font used within the Blogger post composing window by editing your blog's theme HTML or CSS. That part of the interface is controlled by Blogger's platform and is not customizable by individual users.

The good news is that this is purely a visual difference while you're working. Even though the text might not appear in Ubuntu while you're typing or editing, it will display correctly in Ubuntu 12px (or whatever styles you've set in your theme) when you save or publish the post and view it on your actual blog.

So, you don't need to worry that the editor font is different; it won't affect how your readers see your posts.

What's Special About 204?

There's a scene in the first episode of "Prime Target" in which the student asks his supervising professor to find the pattern hidden inside the number (204) that he's written on the blackboard. Initially the professor is relucant but eventually he goes to the blackboard and writes that:$${204}^2={23}^3+{24}^3+{25}^3$$I don't like this series for various reasons and didn't even get through all of the first episode. Before the student wrote on the blackboard, he sitting at the professor's desk and asks the question: "204 is a fascinating number. Don't you find it fascinating?". When the professor tries to get his student back on track (he is reviewing the work that the student has submitted), that's when the student approaches the blackboard and writes the number 204.

Now the professor's response to the student's goading is odd. While the arithmetic of what he's written is correct, the original question asks what pattern is hidden inside the number 204, not its square. Ignoring the number squared and considering just 204, there are a number of possible responses. Let's list them.

• 204 is a sum of all the perfect squares from 1 to 64. In other words:
  $1^2+2^2+\cdots +7^2+8^2=1+4+\cdots +49+64=204$
• 204 is a square pyramidal number: 204 balls may be stacked in a pyramid whose base is an 8 × 8 square. See Figure 1.

Figure 1

• Both 204 and its square are sums of a pair of twin primes:
  $204=101+103\text{ and }{204}^2=41616=20807+20809$
  The only smaller numbers with the same property are 12 and 84. This property does involve the square of the number but the number itself shares the same property.*

There are other properties that are listed in the Wikipedia article but these three stand out. The far more obscure property of the square of the number being equal to the sum of three consecutive integers is shared by 3 and 36 since:$$\begin{array}{rl}{3}^2& =9=0^3+1^3+2^3\\ 36& =6^2=1^3+2^3+3^3\end{array}$$Some of the other properties of 204 are as follows:

• There are exactly 204 ways to place three non-attacking chess queens on a 5 × 5 board.
• There are exactly 204 squares of an infinite chess move that are eight knight's moves from the center.

Anyway, the point is that the choice of the property that the student is thinking of and the professor writes on the board is an odd choice.

* OEIS A213784  Numbers $k$ such that both $k$ and $k^2$ are sums of a twin prime pair. The initial members of this sequence are 12, 84, 204, 456, 1140, 5424, 10044, 11004, 13656, 17940, 27804, 36576.

The Callippic Cycle of 27759 Days

On the 2nd of April 2025, I'm 27758 days old and the connection of this number with the astronomically and astrologically significant $\mathbf{\text{Callippic Cycle}}$ will be revealed later in this blog post. So what is this cycle all about? 

Well let's start with the Metonic Cycle and then we'll get to the Callippic cycle. Here's what Google Gemini had to say about the former.

**************************

The Metonic Cycle: A Cornerstone of Soli-Lunar Calendar Development

The organization of time through calendars has been a fundamental endeavor for human societies, driven by practical necessities such as agricultural planning and the observance of religious rituals, as well as a deeper desire to understand and align with the rhythms of the cosmos.1 Among the earliest forms of calendars were those based on either the solar year, marked by the Earth's revolution around the Sun and the cycle of seasons, or the lunar cycle, defined by the recurring phases of the Moon.3 As civilizations advanced, the need to reconcile these two distinct celestial cycles became apparent, leading to the development of soli-lunar calendars, which sought to integrate both lunar months and the solar year into a unified system.3 A pivotal discovery that significantly advanced the accuracy and functionality of these hybrid calendars was the Metonic cycle.

The Metonic cycle, named after the Athenian astronomer Meton who is credited with its discovery in 432 BCE, is a period of approximately 19 years after which the phases of the Moon recur on roughly the same days of the solar year.8 This cycle arises from the near coincidence that 19 tropical years are almost exactly equal in duration to 235 synodic months.10 More precisely, 19 tropical years amount to approximately 6939.602 days, while 235 synodic months total about 6939.689 days, a difference of a mere 0.087 days, or roughly two hours.10 Meton himself approximated the cycle to a whole number of 6940 days.10 This remarkable near-equality provides a fundamental basis for constructing calendars that can maintain a reasonable alignment with both lunar phases and solar seasons over extended periods.9

Table 1: Duration of the Metonic Cycle

Period	Duration (in days)
19 Tropical Years	≈ 6939.602
235 Synodic Months	≈ 6939.689
Difference	≈ 0.087 (≈ 2 hours)
Meton's Approximation	6940

Soli-lunar calendars, by their very nature, aim to harmonize the monthly cycles of the Moon with the annual progression of the solar year.3 However, a significant challenge in their development stems from the incommensurate lengths of these two astronomical periods. A tropical year, which dictates the seasons, is approximately 365.24 days long 3, while a lunar year, consisting of 12 synodic months, is roughly 354.37 days.4 This difference of about 11 days per year means that a calendar based solely on lunar months will steadily drift relative to the solar year and the seasons.4 In other terms, a solar year is approximately 12.37 lunar months long.10 To address this fundamental mismatch and keep the calendar aligned with the solar year and its seasons, a process called intercalation is necessary, which involves the periodic addition of extra days or, more commonly, an extra lunar month.3 This extra month, often referred to as an embolismic or leap month, helps to reconcile the discrepancy between the lunar and solar cycles.10

The Metonic cycle provides a crucial framework for determining when and how to implement this intercalation in soli-lunar calendars.10 The 19-year cycle offers a predictable pattern for inserting leap months, ensuring that the calendar remains reasonably synchronized with both the lunar phases and the solar year.6 A key element of the Metonic cycle is the rule that within every 19-year period, 7 years will include 13 lunar months, while the remaining 12 years will have the standard 12 lunar months.10 This specific distribution of regular and leap years allows the average length of the calendar year over the 19-year cycle to closely approximate the length of the solar year, thus preventing a significant drift between the calendar and the seasons.6 Ancient civilizations like the Babylonians and the Hebrews effectively utilized the Metonic cycle to structure their intercalation schemes, leading to more accurate and stable soli-lunar calendars.10 Furthermore, the concept of the Golden Number, representing a year's position within the 19-year Metonic cycle, was historically employed as a practical tool for calendar calculations, particularly in identifying the years that would receive an intercalary month.12 Therefore, the Metonic cycle simplified the intricate challenge of creating a calendar that respected both lunar and solar cycles, making it a cornerstone of many ancient timekeeping systems.13

The discovery of the Metonic cycle is attributed to Meton of Athens in 432 BCE, a significant achievement in ancient Greek astronomy.8 Working in collaboration with Euctemon, Meton undertook careful observations of the solstices to determine the length of the tropical year.5 Their calculations revealed an approximate 11-day difference between 12 lunar months and the solar year as they understood it.5 To address this discrepancy in a systematic way, they proposed the 19-year cycle, which incorporated 235 lunar months through a pattern of 12 regular years of 12 months and 7 longer years of 13 months.5 However, evidence suggests that the Babylonians, renowned for their advanced astronomical knowledge, may have been aware of the 19-year cycle and its applications to calendar regulation even before Meton's work.10 Babylonian records indicate the use of a 19-year cycle for intercalation in their lunisolar calendar, possibly dating back to the 5th or 6th century BCE.10 It is conceivable that Meton either independently discovered this cycle or acquired knowledge of it from Babylonian astronomical traditions.13 Intriguingly, interpretations of the ornamentation on the Bronze Age Berlin Gold Hat (circa 1000-800 BC) suggest a potential encoding of a complex counting system that includes the 19-year cycle, hinting at a very early awareness of this astronomical period in Central Europe.10

The Metonic cycle was quickly integrated into the calendrical systems of ancient Greece. Meton himself introduced the cycle to the Attic calendar in 432 BCE, a reform that aimed to improve the synchronization between the lunar months and the solar year.9 Other Greek calendars also adopted or were influenced by this 19-year cycle, demonstrating its perceived accuracy and utility for managing time, especially in relation to religious festivals.14 The high regard for the Metonic cycle in Athens is further evidenced by the practice of inscribing the Golden Number, which indicated the year's position within the cycle, on a pillar in a temple.18 Similarly, the Babylonian calendar, which initially employed more observational and less structured methods of intercalation, formally adopted fixed rules based on the 19-year Metonic cycle around 380 BCE.10 This shift towards a more systematic approach underscores the value placed on the Metonic cycle for achieving a more accurate alignment between lunar and solar time.6 The demonstrated effectiveness of the Metonic cycle led to its adoption or influence in various other ancient cultures, highlighting its significance as a fundamental tool for reconciling solar and lunar timekeeping across different civilizations.10

Numerous ancient soli-lunar calendars utilized the Metonic cycle as a cornerstone of their structure. The Babylonian calendar, for instance, implemented the Metonic cycle to regulate the insertion of an extra month, typically Addaru 2, starting around 499 BCE.10 This systematic intercalation ensured a closer alignment of the lunar calendar with the agricultural seasons, vital for their society.6 The Hebrew calendar remains a prominent example of a modern calendar that has historically and continues to rely on the Metonic cycle.4 It follows a fixed pattern of adding a 13th lunar month (Adar I) in specific years within a 19-year cycle, ensuring that religious holidays like Passover remain aligned with the appropriate time of year.10 The Coligny calendar, a Celtic lunisolar calendar from around AD 200, also utilized the Metonic cycle, showcasing its adoption in regions beyond the Mediterranean.10 In ancient Greece, the Attic calendar, reformed by Meton in 432 BC, directly incorporated the 19-year cycle, demonstrating its early application in the very culture where it was discovered.10 Some historical accounts even suggest that the principles of the Metonic cycle may have been applied to the Roman calendar during the reign of Numa Pompilius (715–673 BC), indicating a potentially very early understanding of this near 19-year synchronization.10

Even today, the Metonic cycle's influence can be seen in several calendar systems. The Hebrew calendar, as mentioned, continues to use the Metonic cycle as the basis for its lunisolar structure, ensuring the alignment of religious observances with the seasons.10 The Runic calendar, a perpetual calendar also known as a Rune staff or Runic Almanac, is based on the 19-year Metonic cycle, demonstrating its utility in creating a timekeeping system that doesn't rely on precise astronomical calculations.10 The Small Maḥzor, a 19-year cycle within the Jewish lunisolar calendar system, is similar to the Greek Metonic cycle and underscores the importance of this period in Jewish timekeeping.10 Even the Baháʼí calendar, while primarily solar, incorporates cycles of 19 years, possibly reflecting an appreciation for the historical significance of this astronomical period.10

The Metonic cycle offered several key advantages for the development of soli-lunar timekeeping. It provided a relatively accurate method for synchronizing solar years and lunar months over a 19-year period, a significant achievement for ancient astronomers.10 The cycle also established a predictable pattern for intercalation, making the maintenance and prediction of the calendar more systematic.10 This predictability facilitated long-term planning for agricultural and religious events, contributing to the stability and organization of societies that adopted it.3 Furthermore, compared to the complexities of achieving perfect astronomical synchronization, the Metonic cycle was relatively simple to implement, relying on a straightforward 19-year pattern of leap months.10

Despite its significant contributions, the Metonic cycle is not perfectly accurate. The 19-year period is not an exact match between 19 solar years and 235 synodic months; a small discrepancy of approximately a few hours exists.10 This slight inaccuracy leads to a long-term drift, causing the calendar based on the Metonic cycle to gradually diverge from the actual positions of the Sun and Moon over extended periods.10 For example, the Hebrew calendar, which utilizes the Metonic cycle, experiences a drift of about one day every 216 years.20 This inherent limitation prompted the development of more accurate calendrical cycles, such as the Callippic cycle (76 years), which aimed to correct some of the inaccuracies of the Metonic cycle.10 Additionally, while the Metonic cycle is related to the recurrence of eclipses, it is not precise enough for accurate eclipse prediction; the Saros cycle is more suited for that purpose.10

The Metonic cycle continues to hold relevance in modern times. It plays a role in the calculation of ecclesiastical dates, particularly in determining the date of Easter in the Christian calendar.10 The Golden Number, derived from the Metonic cycle, is still used in these computations.12 In astronomy, the Metonic cycle provides a useful approximation for understanding the long-term patterns of lunar phases relative to the solar year, which can be valuable for research and observation.9 Furthermore, understanding the Metonic cycle is crucial for interpreting historical astronomical records and appreciating the development of early astronomical knowledge.9 Finally, as seen with the Hebrew calendar, the Metonic cycle remains a foundational element in some modern lunisolar calendar systems, demonstrating its enduring utility.10

In conclusion, the Metonic cycle represents a landmark achievement in the history of calendrical science. Its discovery provided a crucial step forward in reconciling the lunar and solar cycles, enabling the development of more accurate and predictable soli-lunar calendars.10 This advancement significantly benefited ancient civilizations by facilitating agricultural planning, religious observances, and societal organization.3 The enduring influence of the Metonic cycle is evident in numerous historical and some contemporary calendar systems, highlighting its lasting impact on our understanding and measurement of time.10 Despite its slight inaccuracies and the subsequent development of more refined methods, the Metonic cycle stands as a testament to the ingenuity of early astronomers and its significance continues to be felt in our ongoing relationship with the cosmos.19

Works cited

1. Calendars and Their History - AstroPixels, accessed on April 1, 2025, https://astropixels.com/main/calendars.html

2. CALENDARS FROM AROUND THE WORLD - Royal Museums Greenwich, accessed on April 1, 2025, https://www.rmg.co.uk/sites/default/files/Calendars-from-around-the-world.pdf

3. Lunisolar Calendar - (Intro to Astronomy) - Vocab, Definition, Explanations | Fiveable, accessed on April 1, 2025, https://library.fiveable.me/key-terms/intro-astronomy/lunisolar-calendar

4. Lunar calendar - Wikipedia, accessed on April 1, 2025, https://en.wikipedia.org/wiki/Lunar_calendar

5. Calendar - Time, Stars, Sun, Moon | Britannica, accessed on April 1, 2025, https://www.britannica.com/science/calendar/Time-determination-by-stars-Sun-and-Moon

6. Lunisolar calendar - Wikipedia, accessed on April 1, 2025, https://en.wikipedia.org/wiki/Lunisolar_calendar

7. Lunisolar calendar | Britannica, accessed on April 1, 2025, https://www.britannica.com/topic/lunisolar-calendar

8. www.britannica.com, accessed on April 1, 2025, https://www.britannica.com/science/Metonic-cycle#:~:text=Metonic%20cycle%2C%20in%20chronology%2C%20a,was%20discovered%20by%20Meton%20(fl.

9. Metonic Cycle - Solar System Astronomy, accessed on April 1, 2025, https://www.tidjma.tn/en/astro/metonic--cycle/

10. Metonic cycle - Wikipedia, accessed on April 1, 2025, https://en.wikipedia.org/wiki/Metonic_cycle

11. Myaamia Lunar Calendar | Published Research Materials - Miami University, accessed on April 1, 2025, https://miamioh.edu/centers-institutes/myaamia-center/research/publications/lunar-calendar.html

12. Metonic cycle | Moon Phases, Lunar Year & Astronomy - Britannica, accessed on April 1, 2025, https://www.britannica.com/science/Metonic-cycle

13. www.physics.unlv.edu, accessed on April 1, 2025, https://www.physics.unlv.edu/~jeffery/astro/moon/metonic_cycle_girl_with_doves.html

14. Metonic Cycle | Bob Moler's Ephemeris Blog, accessed on April 1, 2025, https://bobmoler.wordpress.com/tag/metonic-cycle/

15. ELI5: Can someone explain metonic cycles? And other lunar ..., accessed on April 1, 2025, https://www.reddit.com/r/explainlikeimfive/comments/86wrdj/eli5_can_someone_explain_metonic_cycles_and_other/

16. Meton | Greek astronomer | Britannica, accessed on April 1, 2025, https://www.britannica.com/biography/Meton

17. Lunar Calendar: tied to the phases of the moon, twelve average lunar synodic months totaling about 354, accessed on April 1, 2025, http://www.yorku.ca/bwall/math5400/projects/calendar

18. CATHOLIC ENCYCLOPEDIA: Epact - New Advent, accessed on April 1, 2025, https://www.newadvent.org/cathen/05480b.htm

19. Introduction to Calendars - Astronomical Applications Department, accessed on April 1, 2025, https://aa.usno.navy.mil/faq/calendars

20. Hebrew calendar - Wikipedia, accessed on April 1, 2025, https://en.wikipedia.org/wiki/Hebrew_calendar

21. Metonic Moon | by Brian Koberlein, accessed on April 1, 2025, https://briankoberlein.com/blog/metonic-moon/

22. Chinese Zodiac Signs - Lunar Date Converter - Fabio Nodari, accessed on April 1, 2025, https://www.fabionodariphoto.com/en/chinese-lunar-date-converter/

23. Cycle of Eclipses - Solar System Astronomy - Simply Explained | Home Page, accessed on April 1, 2025, https://www.tidjma.tn/astro/cycle--of--eclipses/

24. Epact | Catholic Answers Encyclopedia, accessed on April 1, 2025, https://www.catholic.com/encyclopedia/epact

25. www.tidjma.tn, accessed on April 1, 2025, https://www.tidjma.tn/en/astro/metonic--cycle/#:~:text=Modern%20Relevance%3A,various%20astronomical%20observations%20and%20research.

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It was in this report that I read the following:

This inherent limitation prompted the development of more accurate calendrical cycles, such as the Callippic cycle (76 years), which aimed to correct some of the inaccuracies of the Metonic cycle.

I'd never heard of the Callippic cycle before so I was quick to ask Gemini about it.

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The Callippic cycle represents a significant step in the development of accurate soli-lunar calendars. Here's a breakdown of its key aspects:

• Improvement on the Metonic Cycle:
  • The Callippic cycle was proposed by the Greek astronomer Callippus around 330 BC.  
  • It aimed to refine the earlier Metonic cycle, which had identified a 19-year period in which lunar phases roughly repeat on the same days of the solar year.   
  • However, the Metonic cycle wasn't perfectly accurate, and Callippus sought to correct its slight errors.
    
• The 76-Year Cycle:
  • Callippus's solution involved a 76-year cycle, which is four times the Metonic cycle.  
  • He observed that the Metonic cycle was slightly too long, so he adjusted it by subtracting one day from the total count of four Metonic cycles.   
  • This resulted in a cycle of 76 years, containing 27,759 days and 940 lunar months (lunations).   
• Accuracy and Significance:
  • The Callippic cycle provided a more accurate approximation of the relationship between solar years and lunar months.
  • This enhanced accuracy was crucial for creating more reliable soli-lunar calendars, which are essential for agricultural planning, religious observances, and other timekeeping needs.
  • It helped to more accurately predict when lunar phases would occur in relation to the solar year.
• Use in Astronomy:
• The Callippic cycle was used by later astronomers, including Ptolemy, who referenced it in his "Almagest."   
• It played a role in astronomical calculations and observations, helping to establish a more precise framework for understanding celestial motions.

In essence, the Callippic cycle represents a notable advancement in the pursuit of accurate timekeeping, contributing to the evolution of soli-lunar calendars and astronomical understanding.

Sources: 

1. Callippic cycle - Wikipedia en.wikipedia.org
2. This Month in Astronomical History: May 2021 aas.org
3. Time, Stars, Sun, Moon - Calendar - Britannica www.britannica.com
4. Callippus, astronomer, fl. 330 BCE | Oxford Classical Dictionary oxfordre.com
5. Callippic cycle - Oxford Reference www.oxfordreference.com

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27759 is the length of the Callippic Cycle

So from Table 1 earlier, we see that the Metonic cycle is approximately 6940 days and so four cycles amounts to 27760. Callippus subtracted 1 day from this to get 27759 as the length of a Callippic Cycle. My day count begins with $\mathbf{\text{ZERO}}$ however and so at the end of my 27758 days, 27759 days have elapsed. So if we cast a chart for 27758 days after I was born at the same time of day and in the same place, we get the chart shown in Figure 1. 

Figure 1

The Moon and Sun are both in the same degrees of the Zodiac as when I was born: the Sun in the 13th of Aries and the Moon in the 30th degree of Taurus. Remarkable.