Twenty Three

With the 23rd November only two days away, I thought that I'd make a post focused simply on this number 23. I know four people who were born on that day. Firstly, I thought I'd look back over my previous posts and see what references to 23 that I could find:

SUM OF CUBES

Most recently I made a post on Sum of Cubes (October 20th 2020) in which I noted that 23 and 239 are the only integers requiring nine positive cubes for their representation. Only 15 integers require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454. All other numbers require seven cubes or less. In the case of 23 we have:$$23=1^3+1^3+1^3+1^3+1^3+1^3+1^3+2^3+2^3$$CYCLIC NUMBERS

In a post on Cyclic Numbers (November 23rd 2019), I observed that if the digital period of \(1/p\) where \(p\) is prime is \(p\)−1, then the digits represent a cyclic number. 23 is such a prime because it has a period of 22:$$\frac{1}{23}=\text{ 0.0434782608695652173913 }$$Multiplying 0434782608695652173913 progressively by 1 to 22 yields all possible cyclic permutations of this number:

0434782608695652173913    multiplication by 1

0869565217391304347826    multiplication by 2

1304347826086956521739    multiplication by 3

1739130434782608695652    multiplication by 4

2173913043478260869565    multiplication by 5

2608695652173913043478    multiplication by 6

3043478260869565217391    multiplication by 7

3478260869565217391304    multiplication by 8

3913043478260869565217    multiplication by 9

4347826086956521739130    multiplication by 10

4782608695652173913043    multiplication by 11

5217391304347826086956    multiplication by 12

5652173913043478260869    multiplication by 13

6086956521739130434782    multiplication by 14

6521739130434782608695    multiplication by 15

6956521739130434782608    multiplication by 16

7391304347826086956521    multiplication by 17

7826086956521739130434    multiplication by 18

8260869565217391304347    multiplication by 19

8695652173913043478260    multiplication by 20

9130434782608695652173    multiplication by 21

9565217391304347826086    multiplication by 22

Those two posts were the only two that I could find that were of interest regarding 23. Now let's look elsewhere. 

BIRTHDAY PARADOX

23 pops up in the birthday paradox where, in a group of 23 (or more) randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday. Here is the explanation (source):

With 23 people we have 253 pairs: \(\frac{23 \: 22}{2} = 253\). The chance of two people having different birthdays is:$$1 - \frac{1}{365} = \frac{364}{365} = .997260$$Makes sense, right? When comparing one person's birthday to another, in 364 out of 365 scenarios they won't match. Fine. But making 253 comparisons and having them all be different is like getting heads 253 times in a row -- you had to dodge "tails" each time. Let's get an approximate solution by pretending birthday comparisons are like coin flips.We use exponents to find the probability:$$\big( \frac{364}{365} \big )^{253} = 0.4995$$Our chance of getting a single miss is pretty high (99.7260%), but when you take that chance hundreds of times, the odds of keeping up that streak drop. Fast.

SPECIAL PRIMES

• Sophie-Germain prime: a prime \(p\) is a Sophie-Germain prime if \(2 \times p+1\) is prime. In the case of 23, we have 23 x 2 + 1 = 47, a prime. 
  
  
• Safe prime: a prime \(p\) is a  safe prime if \( \frac{p-1}{2} \) is prime. In the case of 23, we have \( \frac{23-1}{2} =11\) and 11 is prime. 
  
  
• Cunningham chain: 23 is the next to last member of the first Cunningham chain (a sequence of prime numbers) of the first kind to have five terms (2, 5, 11, 23, 47). 
  
  
• Twin prime: 23 is the smallest odd prime that is not a twin prime.
  
  
• Woodell prime: a Woodell number \(W_n\) is any natural number of the form \( W_{n}=n\cdot 2^{n}-1\) for some natural number \(n\). A Woodell prime is simply a Woodell number that is prime. 23 is the second such prime after 7. The progression is 7, 23, 383, 32212254719, ... so they are not that common.
  
  
• Factorial prime: a factorial prime is a prime number that is one less or one more than a factorial (all factorials > 1 are even). 23 = 4!-1 and so it qualifies.
  
  
• Eisenstein prime: this is a little complicated but here is a Wikipedia link for learning more about them. The initial Eisenstein primes are 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ...
  
  
• Smarandache–Wellin prime: an integer that in a given base is the concatenation of the first \(n\) prime numbers written in that base is called a Smarandache–Wellin number. If the number is prime, it's called a Smarandache–Wellin prime. The initial such primes are 2, 23 and 2357.
  
  
• Sum of primes: the sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
  
  
• Repunit prime: a number whose digits are all 1 is called a repunit and if that number is prime, then it is called a repunit prime. The 23 digit number 11111111111111111111111 is such a prime 

FACTORIALS

The number 23 is the only prime \(p\) such that \(p\)! is \(p\) digits long. 

23! = 25852016738884976640000

In fact 23 is one of only four numbers \(n\) such that \(n\)! is \(n\) digits long. The others are 1, 22, and 24. The number 23! is the smallest pandigital factorial—it contains each digit at least once.

HUMAN GENOME

Human cells (apart from the sex cells) contain 46 chromosomes: 23 from the mother and 23 from the father. The sex cells contain 23 chromosomes. There are$$2^{23}=8,324,608$$ possible combinations of 23 chromosome pairs and thus$$2^{46}=70,368,744,177,664$$ possible combinations when male and female sex cells combine to produce a human.

CHEMISTRY

• Atomic Number: the atomic number is the number of protons in the nucleus and this number uniquely identifies the element. The atomic number of Vanadium is 23 meaning that it has 23 protons in its nucleus. To quote from Wikipedia:

Vanadium is a chemical element with the symbol V and atomic number 23. It is a hard, silvery-grey, malleable transition metal. The elemental metal is rarely found in nature, but once isolated artificially, the formation of an oxide layer (passivation) somewhat stabilizes the free metal against further oxidation.

Figure 1: source

• Atomic Mass Number: the atomic mass number is the total number of protons and neutrons (together known as nucleons) in an atomic nucleus. The stable isotope of Sodium (Na) has an atomic mass number of 23 (11 protons and 12 neutrons). To quote from Wikipedia again:

Sodium is a chemical element with the symbol Na (from Latin "natrium") and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable isotope is \(^{23}\)Na. The free metal does not occur in nature, and must be prepared from compounds. Sodium is the sixth most abundant element in the Earth's crust and exists in numerous minerals such as feldspars, sodalite, and rock salt (NaCl). Many salts of sodium are highly water-soluble: sodium ions have been leached by the action of water from the Earth's minerals over eons, and thus sodium and chlorine are the most common dissolved elements by weight in the oceans.

ASTRONOMY

To quote from this source:

Today, the Earth's axis is tilted 23.5° from the plane of its orbit around the sun. But this tilt changes. During a cycle that averages about 40,000 years, the tilt of the axis varies between 22.1° and 24.5°.  The average of 22.1° and 24.5° is of course 23.3°.

COSMOLOGY

Not that I believe in dark matter but for those that do, it's postulated that 23% of the Universe consists of it. Here is a graph to convince you it's true!

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 There's a lot more that could be said about the number 23 but that will do for now.

Mathematics in Everyday Life

How many times have I opened a box of tissues by removing the elliptical cover on the top of the box? Every time I do it, I'm aware of its elliptical shape but I never paused to consider the resulting ellipse of cardboard that I held in my hand. It would be discarded as rubbish. Today however, I paused and really looked at what I had in my hand (see Figure 1).

Figure 1

There's even a little semi-circular tab on the right that can be depressed to facilitate the removal of the cover. I'd never noticed that before. Turning the cardboard ellipse over reveals blank cardboard on which I marked in the major and minor axes and measured their lengths, to the nearest millimetre (see Figure 2).

Figure 2

These measurements enable calculation of the eccentricity \(e\) of the ellipse and so in this case, with \(a=28\) and \(b=62.5\) where \(a\) and \(b\) are the lengths of the semi-minor and semi-major axes respectively, we have:$$e=\sqrt {1-\frac{a^2}{b^2}}=\sqrt {1-\frac{28^2}{62.5^2}} \approx 0.894$$This of course is highly elliptical, especially if it's compared with the eccentricities of the planets of the solar system (see Figure 3).

Figure 3

As can be seen in Figure 3, Mercury and Pluto have the most eccentric orbits but much less eccentric than my cardboard ellipse. The other planets have elliptical orbits that would be hard to distinguish from circles if their proportions were displayed on a cardboard cut-out similar to that shown in Figure 2. Coincidentally, there is a centaur with an eccentricity of 0.894 as the table shown in Figure 4 reveals. The academic paper that the table was taken from is quite an interesting but I won't go into here but this is the link, the same as the one shown in Figure 4.

Figure 4

As explained in Figure 4, centaurs are planetesimals with perihelia (closest distance to the Sun) exterior to the orbit of Jupiter and aphelia (farthest distance from the Sun) interior to the orbit of Neptune. The most famous of the centaurs in Chiron, the first to be discovered in 1977 but the somewhat less famous C/2012 H2 (McNaught) does have an orbit that exactly matches that of the cardboard ellipse shown in Figures 1 and 2. Figure 5 provides a little more information about this object.

Figure 5

To calculate the length \(F\) from the centre of the ellipse to the two foci, the following formula can be used involving once again the lengths of the semi-minor and semi-major axes:$$F=\sqrt{b^2-a^2}=\sqrt{62.5^2-28^2} \approx 55.9$$These foci for the cardboard ellipse are shown in Figure 6.

Figure 6

The mathematics in this post is very basic but that was my intention. Though basic, the shape of the cardboard ellipse is nonetheless reflected in the shape of a particular centaur's orbit and it's pretty cool to find a connection between an everyday household item and the solar system in which we are immersed.