23 (twenty-three) is the natural number following 22 and preceding 24. It is a prime number.
Contents
In mathematics
Prime number properties
Twenty-three is the ninth prime number and the smallest odd prime that is not a twin prime. It forms a cousin prime pair with 19 and a sexy prime pair with both 17 and 29. It is the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the next to last member of the first Cunningham chain of the first kind (2, 5, 11, 23, 47). It is the sum of the prime factors of the second set of consecutive discrete semiprimes, (21, 22). 23 is the smallest odd prime to be a highly cototient number, as the solution to
x
−
ϕ
(
x
)
{\displaystyle x-\phi (x)}
for the integers 95, 119, 143, and 529.
23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime, It is the fifth factorial prime, and since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime. 23 is the second Woodall prime, and an Eisenstein prime with no imaginary part and real part of the form
3
n
−
1.
{\displaystyle 3n-1.}
It is the fifth Sophie Germain prime and the fourth safe prime.
23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down. 23 is the smallest prime
p
{\displaystyle p}
such that the largest consecutive pair of
p
{\displaystyle p}
-smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of
(
p
−
1
)
{\displaystyle (p-1)}
-smooth numbers.
The sum of the first nine primes up to 23 is a square:
2
+
3
+
⋯
+
23
=
100
=
10
2
{\displaystyle 2+3+\dots +23=100=10^{2}}
and the sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
The first twenty-three odd prime numbers (between 3 and 89 inclusive), are all cluster primes
p
{\displaystyle p}
such that every even positive integer
k
≤
p
−
3
{\displaystyle k\leq p-3}
can be written as the sum of two prime numbers that do not exceed
p
{\displaystyle p}
.
The twenty-third permutable prime in decimal
R
19
{\displaystyle R_{19}}
is also the second to be a prime repunit (after
R
2
{\displaystyle R_{2}}
), followed by
R
23
{\displaystyle R_{23}}
and
R
1031
{\displaystyle R_{1031}}
.
In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713).
Other mathematical properties
The twenty-third highly composite number 20,160 is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers. Otherwise,
46
=
23
×
2
{\displaystyle 46=23\times 2}
is the largest even number that is not the sum of two abundant numbers.
23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem.
There are 23 trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees.
The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae.
23 is a happy number.
23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.
According to the birthday paradox, 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.
A related coincidence is that 365 times the natural logarithm of 2, approximately 252.999, is very close to the number of pairs of 23 items and 22nd triangular number, 253.
Mersenne numbers
The first Mersenne number of the form
2
n
−
1
{\displaystyle 2^{n}-1}
that does not yield a prime number when inputting a prime exponent is
2047
=
23
×
89
,
{\displaystyle 2047=23\times 89,}
with
n
=
11.
{\displaystyle n=11.}
On the other hand, the second composite Mersenne number contains an exponent
n
{\displaystyle n}
of twenty-three:
M
23
=
2
23
−
1
=
8
In geometry
The Leech lattice Λ24 is a 24-dimensional lattice through which 23 other positive definite even unimodular Niemeier lattices of rank 24 are built, and vice-versa. Λ24 represents the solution to the kissing number in 24 dimensions as the precise lattice structure for the maximum number of spheres that can fill 24-dimensional space without overlapping, equal to 196,560 spheres. These 23 Niemeier lattices are located at deep holes of radii √2 in lattice points around its automorphism group, Conway group
C
0
{\displaystyle \mathbb {C} _{0}}
. The Leech lattice can be constructed in various ways, which include:
By means of a matrix of the form
(
I
a
H
/
2
H
/
2
I
b
)
{\displaystyle \scriptstyle {\begin{pmatrix}Ia&H/2\\H/2&Ib\end{pmatrix}}}
where
I
{\displaystyle I}
is the identity matrix and
H
{\displaystyle H}
In religion
In Biblical numerology, it is associated with Psalm 23, also known as the Shepherd Psalm. It is possibly the most quoted and best known Psalm.
Principia Discordia, the sacred text of Discordianism, holds that 23 (along with the discordian prime 5) is one of the sacred numbers of Eris, goddess of discord.
In popular culture
Film and television
In the TV series Lost, 23 is one of the 6 reoccurring numbers (4, 8, 15, 16, 23, 42) that appear frequently throughout the show.
Other fields
23 skidoo (phrase) (sometimes 23 skiddoo) is an American slang phrase popularized during the early 20th century. 23 skidoo has been described as "perhaps the first truly national fad expression and one of the most popular fad expressions to appear in the U.S".
The 23 enigma, proposed by William S. Burroughs, plays a prominent role in the plot of the Illuminatus! Trilogy by Robert Shea and Robert Anton Wilson.
The Number 23 is a 2007 film starring Jim Carrey about a man who becomes obsessed with the 23 enigma.
The number 23 is used a lot throughout the visuals and music by the band Gorillaz, who have even devoted a whole page of their autobiography Rise Of The Ogre to the 23 enigma theory.
Human cells have 23 pairs of chromosomes (22 pairs of autosomes and one pair of sex chromosomes).
