28003: A Lesson Learned

One of the limitations of the free version of the Airtable online database is that it only allows about 1000 records per database. Once the limit is reached, you need to create a new database to accommodate your needs. I've been using Airtable for years now and I am into my third database:

• Diurnal Age Part 1
• 1198 records (over the limit)
• started on March 17th 2017
• closed on June 26th 2021
  
  
• Diurnal Age Part 2
• 923 records
• started on June 27th 2021
• closed on January 2nd 2024
  
  
• Diurnal Age Part 3
• 704 records currently
• started on January 3rd 2024
  
  

This is an impressive combined database containing about 3000 records. However, the records are in three disconnected databases and once I move on to a new database, I've tended to ignore the earlier databases. However, I realised the folly of doing this when confronted with something interesting to say about the number associated with my diurnal age today: \( \textbf{28003}\).

It's not dark yet but it's getting there

The OEIS, Mathematical Meanderings (my Mathematics blog), Bespoken for Sequences and Diurnal Age Part 3 (in short all my usual sources) had nothing useful to contribute. It then occurred to me to look back at my Diurnal Age Part 2 database and I was pleasantly surprised at what I found therein. Let's begin:

• \( \textbf{28003} \) is a member of OEIS A179248: numbers that have 8 terms in their Zeckendorf representation: [3, 8, 89, 233, 610, 2584, 6765, 17711]. There is in face a cluster of sequence members nearby, namely 27800, 27802, 27804, 27807, 27808, 27809 and 27811.
  
  
• \( \textbf{28003}\) is a member of OEIS A157344: semiprimes that are the product of two distinct Sophie Germain primes. Approximately 2.52% of numbers in the range up to 40,000 satisfy this criterion. Here the primes are 41 and 683 and we have:
• 41 x 2 + 1 = 83 which is prime
• 683 x 2 + 1 = 1367 which is prime
  
  
• \( \textbf{28003}\) is a member of OEIS A082957: numbers \(n\) such that \( \sigma(2n) < \sigma(2n+1) \). These numbers total 5.52% in the range up to 40,000. There are 2301 primes and 3169 composites among the 5470 first terms. Here we have:
• \( \sigma(2 \times 28003) = \sigma(56006) = 86184\)
• \( \sigma(2 \times 28003 + 1) = \sigma(56007) = 94848\)
  
  
• \( \textbf{28003} \) is a member of OEIS A134252: positions of 2 after decimal point in decimal expansion of \( \frac{1}{\pi}\). 

Diurnal Age Part 1 yielded no results but as can be seen Diurnal Age Part 2 revealed membership of \( \textbf{28003}\) in four different OEIS sequences.

Trouble With Numbers Aplenty

Unfortunately one of the mathematical sites that I use on a daily basis has been difficult to access recently and today it refuses to open at all. Figure 1 shows the message that appears.

Figure 1

My daily tweet always contains a link to Numbers Aplenty as well as a screenshot. Figure 2 shows yesterday's tweet.

Figure 2

Similarly my AirTable database always contains a link to Numbers Aplenty. I put the link up for today even though it doesn't currently work. See Figure 3.

Figure 3

It would be most unfortunate if the site were to go offline permanently. Let's hope that doesn't happen. I put an update in this post once things are back to normal.

POSTSCRIPT: happily, the site now seems back to normal as very late in the evening of October 1st 2024 (Melbourne time).

The Number Plate Game

Figure 1

Most number plates in Indonesia display a four digit number as shown in Figure 1 where the number is 1234. A game I like to play as I drive around or am driven around is to try to make a digit equation from these four digits. My rules are that only the four operations of addition, subtraction, multiplication and division are allowed together with exponentiation and brackets. Concatenation of digits is not allowed. Division can be of two types:

• a digit divided into the next digit e.g. 2 | 4 = 2 which reads 2 divided into 4
• a digit divided by the next digit e.g. 4 / 2 = 2 which reads 4 divided by 2

For the 1234 in Figure 1, the digit equation is very straightforward:$$ \begin{align}  -1 + 2 &= -3 + 4 \\ 1 &=1 \end{align}$$Theoretically there are \(10^4 = 10000 \) possible numbers ranging from 0000 to 9999. Any numbers with at least two zeroes will form a digit equation. Take 1009 as an example:$$ \begin{align} 1 \times 0 &= 0 \times 9 \\ 0 &=0 \end{align} $$If the digit 1 appears at least twice in the first three digits, then a digit equation is always possible. Take 1814 as an example:$$\begin{align} 1^8 &= 1^4 \\ 1 &=1 \end{align}$$Two consecutive digits are often helpful because they can be collapsed to a 1. Take 6599 as an example:$$\begin{align} 6 - 5 &= 9 \, /  \,9 \\ 1 &=1 \end{align}$$One or more zeroes are also helpful because raising any digit to the zero power produces a 1. Take 9032 as an example:$$\begin{align} 9^0 &= 3-2 \\ 1 &=1 \end{align}$$Brackets also prove very useful as shown in the example of 4894 where we bracket the 8 and 9 and make the 8 negative:$$\begin{align} 4^{-8+9} &= 4 \\ 4^1 &=4 \\ 4&=4  \end{align}$$Of course, forming a digit equation can require a little trickery. Take 3649 as an example:$$\begin{align} 3^{ 6-4}&= 9 \\ 3^2 &=9 \\ 9 &= 9 \end{align}$$Sometimes a digit equation is not possible according to the rules imposed. Take 3637 as an example. I can't see how to make a digit equation out of those four digits.

On the other hand, it's often possible to form digit equations in several different ways. Take our original 1234 as an example where another representations is possible:$$\begin{align} 1^2 &= -3+4 \\ 1 &=1 \end{align}$$It would be an interesting exercise to work out digit equations for all the numbers between 0000 and 9999 and to make a note of those numbers where an equation is not possible. In fact I've began this process in an Airtable database (link) and it would appear that 0124 is the first four digit number to thwart the creation of a digit equation. I'll keep working on the database and make additions to this post when I discover numbers of interest. It will of course take some time to complete.

See post titled Forming Equations from the Digits of a Number from the 14th of March 2024.

ADDENDUM: July 10th 2024

A variation of this game would be to see if one can get the numbers to total zero. This is very similar to the game already described and every solution to that game becomes a solution to this game because \(a=b \implies a-b=0\). This new, let's call it "zero game", allows for additional numbers plates containing the digit zero to qualify. For example 2470:$$2470 \rightarrow 2 \times 4 \times 7 \times 0 = 0$$Under the old rules this doesn't qualify but here the product of the digits is zero so it does. Another example would be 6772:$$6772 \rightarrow 6 \times  (7 - 7) \times 2 = 0$$Under the old rules, a digit equation could not be formed from these digits but here it can.

Diurnal Age Meets Conway's Game Of Life

 I've written about Conway's Game of Life in two recent posts:

• The Game of Life on Saturday, the 16th of December 2023 and

• 108 Meets Conway's Game Of Life on Friday, the 9th February 2024

I've been playing around with an app called "Life" on my iPhone that allows the game to be run but I prefer on browser-based app that I can access from my laptop. To that end, I've been playing around with one of three websites recommended by Gemini:

• Conway's Game of Life (Golly edition): https://conwaylife.com/

This website utilizes the popular "Golly" simulation software, offering advanced features like pattern libraries, scripting, and different grid geometries. You can save and export your simulations in various formats.

• Game of Life Online: https://playgameoflife.com/

This website allows you to draw patterns directly on the grid with an intuitive interface. While it lacks advanced features, it's great for quick visualizations and sharing creations.

So in this post, I'm looking at the first of the recommendations and playing around with a new idea. I want to investigate how the number associated with my diurnal age behaves under the Game of Life rules. The number for today, 27346, is shown in Figure 1. All the digits from 0 to 9 can be created using a 3 x 5 pixel grid, the smallest possible size.

Figure 1

The rules lead, after 125 steps, to the image shown in Figure 2:

Figure 2

What would be interesting to keep track of are the number of steps required to reach a stable state. It's clear that the stable states arising from numbers are not unique. For example 16161 will end up the same as 19191 if we don't regard mirror images, rotations and reflections as different. However, most numbers should result in stable states that are different from one another. I can attach images of these stable states to my Airtable database. 

I'll explore the other two Gemini recommendations later. Any particularly interesting stable states or record number of steps arising from these diurnal age investigations can be the subject of future posts. In the case of 27346, we can say that the stable state consists of five blocks (the simplest still life) and one hive or beehive (the second most common still life).

 

This ongoing, daily exercise is a great way to deepen ones understanding of a topic. It was only through my adherence to the investigation of the number associated with my diurnal age that I widened and deepened my understanding of number theory. It's a great maxim: once a day but everyday and can be and should be applied to more aspects of my daily life.

Measuring Dartsmanship

Recently I've taken to recording how many throws it takes me to complete a game of Round the World in darts. The game is quite simple: you must score a 1 before moving on to the 2, once the 2 is completed you can move on 3 and so on around the board before finishing on the red bullseye or the green ring around it. The minimum number of throws required for this feat is 21.

I decided to measure the efficiency of my score by dividing it into 21 and expressing this as a fraction. Thus: $$ \text{efficiency }=\frac{21}{\text{score}} \times 100$$Figure 1 shows a graph of the resultant efficiencies for scores ranging from 105 to 21.

Figure 1: permalink

Figure 2 shows a table of selected scores and their associated efficiencies (rounded to the nearest whole number).

Figure 2: permalink

As can be seen, it becomes increasingly difficult to achieve an efficiency close to 100%. For example, 22 scores 95% and 21 scores 91% but all other scores are under 90%. I'm recording these results in a newly created AirTable database. See Figure 3.

Figure 3

I'm keeping track of the number of throws using a counter on my iPhone. See Figure 4.

Figure 4

There is a defect of sorts in this way of measuring efficiency because it supposes that 21 steps required to finish are all equal. Indeed from 1 to 20 they are but the final bullseye and green ring have a combined area that is considerably smaller than the numbered sectors. One might complete steps 1 to 20 with 20 throws and then spend ten more throws before hitting the central area of the dartboard. The final score of 30 with result efficiency of 70% doesn't fully reward the extraordinary skill required to progress from 1 to 20 in only 20 throws.

The bull's-eye has an outerbull area (also know as the single bull, which scores 25) and an inner bull (also known as a double bull's-eye, which scores 50). The circular scoring area of the standard dartboard has a diameter of 34" and the bull's-eye has a diameter of 3". So this means that each numbered sector has an area of 45.04 square inches and the bull's-eye has an area of 7.069 square inches which is thus more than six times smaller. Hitting the bull's-eye in a single throw ought to be rewarded more than hitting one of the numbered sectors in a single throw.

Hitting the bull's-eye is equivalent to hitting six numbered sectors in succession. The numbers should range from 1 --> 20 and then from 21 --> 26 but 26 is difficult to work with. Let's go with 21 --> 25 so that the numbered sectors count for 80% and the bull's-eye 20%. If somebody hits the numbers 1 to 20 in twenty throws, they are assured of an 80% score. A formula then involves two statistics, \(x\) and \(y\) where the former represents the throws taken to traverse 1 to 20 and the latter represents the throws needed to hit the bull's-eye. The formula thus becomes:$$\text{efficiency }=\frac{20}{x} \times 80 + \frac{1}{y} \times 20$$This is a fairer estimate of efficiency that doesn't unduly penalise somebody for having difficulty hitting the bull's-eye. I've changed my AirTable database to reflect these changes.

ADDENDUM: March 6th 2024

There's a major problem with this final efficiency formula that I came up with and I only noticed it today. Recently, on February 26th, I achieved an efficiency of 62% after scoring 41 in the 1 to 20 section and 1 in the bull's-eye:$$ \text{efficiency } =\frac{20}{40} \times 80 + \frac{1}{1} \times 20  \approx 62.0 \%$$Today I needed 41 for the 1 to 20 section but needed two throws to get the bull's-eye. However, I was shocked to see that my efficiency was 10% less as the result of the calculation:$$ \text{efficiency } =\frac{20}{41} \times 80 + \frac{1}{2} \times 20  \approx 52.0 \%$$This is clearly not reasonable but the problem only emerged as the consistency of my dart throwing improved.

For the time being, I'll revert to my original formula:$$ \text{efficiency }=\frac{21}{\text{score}} \times 100$$This then produces more reasonable results:$$ \text{efficiency }=\frac{21}{41} \times 100 \approx 51.2 \%$$$$ \text{efficiency }=\frac{21}{43} \times 100 \approx 48.8 \%$$This formula is far from perfect but it will have to do for the time being until I come up with something better.

Special Numbers, Special Dates

On the 26th January 2023, I posted about Turning Dates Into Numbers and made the observation that the 10th February 2023 was a special day because the number associated with this date (20230210) had the interesting property that it was the only date in the year that produced a number that had seven distinct prime factors:

20230210 = 2 x 5 x 7 x 11 x 13 x 43 x 47

It turns out that this number has another interesting property in that the sum of its digits equals the last two digits (representing the day of the month). In fact, it begins a run of ten consecutive numbers that end with 20230219 (the 19th February 2023) ... permalink:

20230210, 20230211, 20230212, 20230213, 20230214, 20230215, 20230216, 20230217, 20230218, 20230219

Another run of ten similar numbers occurs in November, starting on the 10th and ending on the 19th.

20231110, 20231111, 20231112, 20231113, 20231114, 20231115, 20231116, 20231117, 20231118, 20231119

For many, many years to come, the sum of the digits of the numbers associated with the dates will always contain two digits. Today is the 15th February 2023 and so we are into the second half of the first-mentioned run.

This number can be expressed as a sum of three cubes
See Figure 2

Similar runs will occur every year but the dates will differ naturally. For example, in 2024, the first run of numbers will begin on 11th January 2024 with 20240111 and end on 19th January 2024 with 20240119. 

In my previous post Turning Dates Into Numbers I looked at what numbers were prime, semiprime and so on. In this post I'll turn my attention to cubes and see what numbers can be expressed as the sum of two cubes and as a sum of three cubes. 

It turns out that there is only one number in the entire year that can be expressed as a sum of two cubes and that is \( 20231020 = 69^3 + 271^3\). This corresponds to the 20th October 2023.

The sums of three cubes are more common. The complete list is shown in Figure 2, arranged in chronological order with 20th January being the first date and 26th December the last.

Figure 2: permalink

It can be noted that some dates produce numbers that can be expressed as sums of three cubes in more than one way. Specifically, 20230426 (two ways), 20230523 (two ways), 20230722 (two ways), 20230904 (two ways), 20230911 (three ways), 20231028 (two ways) and 20231226 (two ways). Thus the 11th September (the famous 911 in United States date format) produces the only number (20230911) that can be expressed as a sum of three cubes in three ways.

A History of Tweets

Each day I tweet about the number associated with my diurnal age but I rarely look back at any of these tweets, especially now that they just represent a summary of a far more detailed analysis that I have stored on my AirTable databases (Diurnal Age Part 1 and Diurnal Age Part 2). 

However, this was not always the case and intially my daily tweet was the only record I retained of my analysis. I discovered a site (https://www.allmytweets.net/) that allows you to easily look back at all your previous tweets and it was interesting to look back at my very first tweet on April 30th 2015. See Figure 1.

Figure 1

Short and sweet but this was back in the days where only 140 characters were allowed. It was only on November 8th 2017 that Twitter increased the limit to 280 characters. This is the OEIS sequence to which the tweet refers:

A109724

Sum of the first \(n^2\) primes.

The initial members of the sequence are:

0, 2, 17, 100, 381, 1060, 2427, 4888, 8893, 15116, 24133, 36888, 54169, 77136, 106733, 144526, 191755, 249748, 320705, 406048, 507825, 627294, 768373, 931686, 1119887, 1336090, 1583293, 1864190, 2180741, 2536646, 2935471, 3380980

In the case of 24133, \(n\) has the value 10. I didn't tweet the next day but did for 24135 and then I missed another day and tweeted for 24137. After that I kept tweeting for every day without fail until the day of this post. The tweets were not exclusively mathematical in those days. It was only on August 6th 2016 (24588) that the tweets became solely mathematical.

I'll still keep tweeting, old habits die hard, even though my AirTable database and my mathematical blog (Mathematical Meanderings) are the foci of my mathematical analysis these days.

Of course I was counting days long before I started tweeting about them. Here is a graphic I created to celebrate being 22222 days old. This occurred on Thursday, February 4th 2010!