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.
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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.
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