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# 23

Kristen Eller ©2002 |

The smallest prime whose reversal is a power: 32 = 2^{5}. [Trigg]

23 is the smallest prime for which the sum of the squares of its digits is also an odd prime. [Trotter]

The floor function of *e*^{π} = 23.

The suicidal bomber has seat number 23 in the film *Airport* (1970).

The smallest prime number that is not the sum of two Ulam numbers. The (standard) Ulam numbers start with 1 and 2; then the subsequent terms are the smallest numbers that can be expressed in just one way as the sum of two distinct earlier terms. The sequence begins 1, 2, 3, 4, 6, 8, 11, 13, .... [Guy]

23^{0} + 23^{1} + 23^{2} + 23^{3} + 23^{5} = 6449063 and the sum of 6 + 4 + 4 + 9 + 0 + 6 + 3 can be
written as 3 * 2 + 2 * 2 + 2 * 2 + 3 * 3 + 0 + 3 * 2 + 3 = 32, the reversal of 23. [Gevisier]

23 is the only prime number p such that p! is p digits long. [Gupta]

The smallest odd isolated prime, i.e., not an element of a set of twin primes. [Francis]

"Hilbert's problems" are a list of 23 problems put forth by David Hilbert in the year 1900. Note the number of words in the following quote attributed to him: "If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?" [Beedassy]

The "23 Enigma" is a belief that the number 23 is of particular or unusual significance.

Every positive integer is the sum of at most nine cubes. Only 23 and 239 require all nine! Waring's problem (1770) asks: if given k, is there some number g(k) for which every positive integer can be written as the sum g(k) (or fewer) kth powers? For example, g(3) = 9 and g(4) = 19. In 1909, Hilbert proved the answer was yes.

Psi (y) is the 23rd letter of the Greek alphabet. [Roberts]

*Homo sapiens* have 23 pairs of chromosomes.

(3^{5} + 5^{3})/(3+5 + 5+3) = 23. [Kulsha]

The largest prime with its least quadratic non-residue greater than sqrt(*p*). [Nagell]

The smallest prime that is both preceded and followed by a prime number of successive composites. [Beedassy]

Cyclotomic rings are formed by adjoining the *n* roots of unity to the integers. Montgomery and Uchida independently showed that the
first of these rings to have class number greater than one is formed by adjoining the 23rd roots of unity. [Bonan]

There are only 23 odd abundant numbers below 10^{4}.

23 is the smallest prime of the form 10**p* + 3 that is not the sum of
two squares, where *p* is prime. [Murton]

11111 * 1111^{11} + 11111111111 + 1 (23 ones) is prime. [Fougeron]

2^{n} + 3^{n} is prime for *n* = 0, 1 and 2. [Trotter]

Sherlock Holmes and Doctor Watson lived at 221b Baker Street for a period of 23 years. [Haga]

There are 23 discs in the human spine. [McCranie]

The top speed of an American crow is 23 miles per hour. [Stidham]

The smallest prime such that the product of its digits is a perfect number. [Kumar]

23! is the least factorial in which the digits 0 through 9 appear at least once. [Das]

W is the 23rd letter of the modern English alphabet. Have you ever noticed that it has 2 points down and 3 points up? W is also situated below the letters 2 and 3 on a "qwerty" keyboard!

23 is the only prime of the form p*q + p + q and p*q - p - q, where p and q are two successive primes (3*5 + 3 + 5 = 5*7 - 5 - 7 = 23). [Teofilatto]

There are 23 definitions in Book I of Euclid's *Elements*.

π(23) = 3^{2}. [Caldwell]

23 = 5 + 7 + 11. Do you see the first five consecutive primes? [La Haye]

23 is the smallest prime number with consecutive digits. [Capelle]

23 = 3 times the third prime + twice the second prime + once the first prime. [Murthy]

2! + 3! = 2^{3}. [Murthy]

23 = - (2^{2} - 3^{3}). [Murthy]

Thrice three and twenty for pleasure true and plenty. [Croll]

The smallest prime of form p^{p} - q^{q}, where p and q are primes. [Murthy]

Smallest prime p such that 90*p+11, 90*p+13, 90*p+17, 90*p+19 are all primes. [Russo]

23 is the smallest prime number that is equal to the product plus sum of twin primes, i.e., 3*5 + (3+5) = 23. [Wagler]

At the height of his career, Professor John F. Nash, Jr., interrupted a lecture to announce that a photo of Pope John XXIII on the cover of LIFE magazine was actually himself (Nash) in disguise and that he knew this because 23 was his favorite prime number. [Hageman]

The smallest prime p that divides the number of digits of p!. [Russo]

The largest integer that cannot be expressed as the sum of two nonsquarefree numbers. A number is nonsquarefree (or non-squarefree) if it contains at least one square in its prime factorization. [Rupinski]

The sum of the square roots of the first 23 primes is very near the 32nd prime. [Rupinski]

In a room of just 23 people, a greater than 50% chance exists that two of the people will share a common birthday.

Archbishop Ussher (1581-1656), Primate of All Ireland, argued that the world was created on October 23, 4004 B.C. [Delval]

In the Charles Dickens novel *A Tale of Two Cities*, each beheading was counted down. Sidney Carton, the insolent, indifferent, and alcoholic attorney, was guillotine victim number 23. [Nunes]

"Twenty-three, skidoo!" is an American catch phrase of unknown origin.

Julius Caesar was stabbed 23 times when he was assassinated.

23 is the second Woodall prime.

"The Immortal Game" of chess, played in London on June 21, 1851, between Adolf Anderssen and Lionel Kieseritzky, ended after Anderssen sacrificed a bishop, two rooks, and his queen to deliver checkmate in 23 moves. Below is a record of this in algebraic notation:

1. e4 e5 2. f4 exf4 3. Bc4 Qh4+ 4. Kf1 b5 5. Bxb5 Nf6 6. Nf3 Qh6 7. d3 Nh5 8. Nh4 Qg5 9. Nf5 c6 10. g4 Nf6 11. Rg1 cxb5 12. h4 Qg6 13. h5 Qg5 14. Qf3 Ng8 15. Bxf4 Qf6 16. Nc3 Bc5 17. Nd5 Qxb2 18. Bd6 Bxg1 19. e5 Qxa1+ 20. Ke2 Na6 21. Nxg7+ Kd8 22. Qf6+ Nxf6 23. Be7 mate

A factorial expression involving the digits of 23 is 1! + (2! + 2!) + (3! + 3! + 3!). [Capelle]

There are exactly 23 nonisomorphic trees with 2^3 vertices. [Patterson]

The pseudoscientific "biorhythm theory" claims everyone has a 23-day physical cycle that influences their general physical condition.

One 2 and one 3 (23) divides two 2's and three 3's (22333). [De Geest]

2^23 + 3^23 + 2*3 is prime. [Laurv]

There are 23 primes which can not be written as a sum of (nonunit) cubes. Note that 23 is the largest prime which cannot be written as a sum of (nonunit) squares. [Hartley]

The world's fastest human can average 23 miles/hour in the 100 meter dash. [Adrion]

23 differs from its reversal by 3^2. [Markowitz]

The largest of the six primes which are uniquely expressible as the sum of at most 4 squares. The others are 2, 3, 5, 7, and 11. [Rupinski]

The word hell is found only 23 times in the King James Version of the New Testament, and most of those are by Jesus. [Van Empe]

The Sundial Bridge, a footbridge over the Sacramento River at Redding, California, is just short of 23 feet wide. [Haga]

23 is the smaller prime factor of 2^11-1, the smallest composite Mersenne number with prime exponent. [Terr]

The smallest Pillai prime.

23 is the first happy number that is not of the form 3n+1. [Axoy]

23 is the only known number p where R_{p} is a repunit prime, 10^p-p is prime, and p is a prime factor of a Mersenne number with repunit exponent. I.e., 23 is factor of 2^11-1. [Luhn]

23*3-32*2 = (3+2)*(3-2). [Losnak]

The primes up to 23 and the squares of primes up to 23 separated by zeros form primes. [Gallardo]

Dr. Pepper is a unique blend of 23 secret flavors. [Dowdy]

23 is 2 (mod 3) and 3 (mod 2). [Losnak]

23 is the smallest and only prime of the form 2x^2 - y^2 and 3x^2 - y^2, where x and y are two consecutive numbers. [Teofilatto]

23 is the smallest prime equal to the sum of three distinct primes in two ways: 5+7+11 = 3+7+13 = 23. [Teofilatto]

To get the largest 23-digit prime one only needs to subtract 23 from 10^23. [De Geest]

23 = (2^2 + 3^3) - (2! + 3!). [Teofilatto]

No one really knows exactly when the great English poet and playwright William Shakespeare (1564-1616) was born, but, traditionally St. George's Day (April 23) has been his accepted birthday. However, we do know that he died on this date. [Dowdy]

The only prime in a sequence of six numbers that regularly occurred in the first season of the television series *Lost*. [Hartley]

The smaller prime of the smallest set of two consecutive multidigit primes (23, 29) whose sum of digits (5, 11) is another set of two distinct primes. [Gupta]

The first number with a prime number of digits, all digits prime, and the sum of all digits being prime. [Dickman]

23 is the largest row of Pascal's triangle all of whose entries are square-free (or 1). [Beedassy]

The sum of first 23 primes is 874 (a multiple of 23). Note that 874 = 23 * 38 and the 23rd prime is 83. [Poo Sung]

If you were able to fold (doubling each time) a sheet of paper (0.01 in) 23 times, it would become over a mile thick. [Rogowski]

October 23 is Mole Day. It is celebrated among chemists in North America from 6:02 A.M. to 6:02 P.M. on 10/23 each year in honor of Avogadro's number, which is approximately 6.02 * 10^23. [Pritesh]

David Bloomberg (Bryan Greenberg) is a penniless
23-year-old painter when he meets the fourteen-years-older
Rafi Gardet (Uma Thurman) in the movie *Prime*. [Opao]

The only possible binary "perfect code" that corrects two or more errors is the three error correcting Golay code of length 23. [Kerdock]

23 is the smallest prime with a composite number of 1s in its binary expansion. [Keith]

In the movie "Support Your Local Gunfighter," starring James Garner, the protagonist has an obsessive compulsion to risk all his money in roulette on the number 23. It causes him to consistently lose any significant money he amasses. He says of the number, "It pulls at me." [Wynn]

The smallest non-reflectable non-strobogrammatic non-invertible non-palindromic non-reversible prime number. [Capelle]

The number of characters in the interjection : "Yeah! Hints on the primes" (a most fitting anagram for "The Riemann Hypothesis" due to V. Rabin). [Beedassy]

23 = 2^3 + 3^2 + 2*3. [Hollesen]

The sum of the digits of 23^^{2+3} is the next prime after 23. [Silva]

Joe the Plumber (Samuel Joseph Wurzelbacher) was mentioned by John McCain and Barack Obama at least 23 times during the presidential debate of October 15, 2008. [ABC News]

The smallest prime *p* that divides the sum of the first *p* primes. [Beedassy]

The sum of the consecutive primes from 23 to the 23rd prime is a palindromic prime. [Silva]

There are exactly 23 prime-digit primes below one thousand,
and 23 is one of them (more precisely the *n*th,
where *n* is the (prime) sum of its digits). [Beedassy]

There are 23 known natural numbers n such that
2^{n} does not contain the digit
2. Their sum ends with 23, which is one of them. The prime
cases are easy to remember: 2, 3, 23 and ... the 23th
prime! [Capelle]

23 + 2*3, 23 + 2^3, and 23 + 2*3 + 2^3 are the three next primes after 23. [Silva]

The smallest prime which when prefixed to the smallest pandigital string forms another prime: 230123456789. [Beedassy]

23 = 0^{5} + 1^{4} + 2^{3} +
3^{2} + 4^{1} + 5^{0}. [Poo Sung]

The sum of the prime divisors of its two previous numbers. [Silva]

The first prime whose complexity is a prime smaller than itself (i.e., whose representation involving only addition, multiplication, brackets and 1's makes use of a smaller prime number of 1's): {(1+1)*(1+1+1)+1}*(1+1+1)+1+1. [Beedassy]

The largest known prime number that cannot be written as the sum of at most three powerful numbers. [Capelle]

The smallest prime *p* such that a prime is formed both from the ordered and the reverse concatenations of the first noncomposites up to *p*: 123571113171923 ; 231917131175321. [Beedassy]

Professional basketball player LeBron James switched his number 23 to the first perfect number. [Plaster]

The smallest Sophie Germain prime formed by concatenation of two others. [Silva]

Digits [2|3] is the only 2-digit number that is prime in 5(2+3) different base numbers equal or below base (10): 23(10)=23, 23(8)=19, 23(7)=17, 23(5)=13, and 23(4)=11! [Elthon]

23 is the smallest prime of the form 2(3)_{k}. [Green]

The smallest safe prime that equals to the sum of the previous n safe primes (n=3). [Loungrides]

Dwayne "The Rock" Johnson plays Agent 23 in the 2008 movie GET SMART. [Green]

The smallest n-digit prime that yields n others if each of its digits is duplicated (223, 233). [Silva]

The smallest multidigit prime that is simultaneously a left-truncatable prime and right-truncatable prime. [Loungrides]

The smallest prime p that divides the reversal of the next prime after p. The next three pairs of consecutive primes of this form and with this property are: 233 239; 2333 2339; 23333 23339. Are there primes of other forms with this property? [Silva]

There are 23 essentially different ways to color the nine squares of a 3-by-3 checkerboard, such that 5 of the squares are black, and 4 are red. Z = RED, O = BLACK:

OZO OZO ZOO ZZO ZZO ZZO ZOO ZOO ZOZ ZZZ ZZZ ZZO OZZ ZOZ ZOZ OZZ OZO OOO OOO OOO OOO OOO OZO OZO ZZZ ZZZ ZZZ ZZO ZZO ZZO ZZO ZOZ ZOO OOZ ZOZ ZOO OZO OOO OOO OOO OOZ ZOO ZZO ZZO ZZO ZOO OOO OOO OZO ZZO OZZ OOZ OOZ OOO OOZ OOZ OZO ZZZ ZOZ ZOZ ZOZ OOO ZOO OZO OOO ZOO OOZ OOZ ZOZ[Green]

The smallest home prime for composite numbers. [Beedassy]

In the James Bond picture "Goldeneye," Valentin Zukovsky "operates out of Building Number 23 there."

A couple is two people, a prime. But the Guinness World Record for most marriages (and divorces) is 23 by a woman in Indiana. There is also a record for most renewal vows: the prime 83. [Post]

*"PneumoVax 23"* is the trade name for an
anti-*Streptococcus pneumoniae* germ vaccine
containing 23 of the most virulent and prevalent
pneumococcal antigen serotypes. [Beedassy]

Polynomial equation (x^2+3)(x^3+2)=0 does not have integral solutions although is has a solution in Z/pZ for every prime p. [Jevtic]

There are 23 words in David Hilbert's statement on why he didn't try to solve Fermat's last theorem: "Before beginning I should put in three years of intensive study, and I haven't that much time to squander on a probable failure." Quoted in E. T. Bell Mathematics, Queen and Servant of Science (New York 1951). [Post]

The smallest prime p such that neither p+1 nor p-1 is divisible by the sum of the digits of p. [Silva]

Largest known prime partial sum of number of Giuga sequences of length n: 1 + 2 + 3 + 17. [Post]

The only known prime difference between the number of semigroups of order n and the number of groups of order n, for any integer n. For n = 3, there are 24 nonisomorphic semigroups, and a unique (cyclic) group, so 24-1 = 23. [Post]

On the basis that A = 1, B = 2, C = 3 ..., Z = 26, then ODD (23) is odd. Note that mathematically odd + odd = even and ODD (23) + ODD (23) = EVEN (2 * 23). [Homewood]

The smallest prime with equal number of even and odd digits. [Wesolowski]

(23, 29) is the only pair of consecutive double-digit primes p, q, such that the sum of digits of p^5 is equal to q and the sum of digits of q^5 is equal to p, (23^5=6436343 (whose sum of digits is 29) and 29^5=20511149 (whose sum of digits is 23). [Loungrides]

The *Qur'an* was revealed over a 23 year period. [Broschart]

(17, 19, 23) is the only known triplet of multidigit consecutive primes p < q < r such that r|pq-1. [Loungrides]

(23, 29) is the only pair (p, q), of two consecutive double-digit primes (p < q), such that p^2+q^3-/+1, is a twin prime pair, i.e., (24917, 24919). [Loungrides]

sod(M22 exponent) = sod(M28 exponent) = sod(M37 exponent) = 23. [Harrison]

The arithmetical mean of first 23 prime-digit primes is an integer. [Bajpai]

The end of the world may be on 23 April 2018, if a Christian numerologist (David Meade) is to be believed.

The integers from 1 to 23 contain the ten digits a prime number of time each. [Gaydos]

If you add the successive digits of π, then the first prime number total is 23. Note that the final digit added is 9 and 23 is the 9th prime number. [Homewood]

23+2 = (2+3)^2. [Silva]

The "New York Times" Sunday crosswords are sometimes 23 by 23. [Velshi]

How many terms of the series 1 + 1/2 + 1/3 + ... + 1/9 + 1/11 + ... (which excludes all denominators containing 0) is required to produce a sum greater than 23? [Honaker]

23 is the only homosexual prime, i.e., a sexy prime which can appear as its own gap: 23 + 2*3 = 29. [Silva]

(2^9 + 3^8 + 5^7 + 7^6 + 11^5 + 13^4 + 17^3 + 19^2 + 23^1) + 1 is prime. [Wiszowaty]

Asylum Patient No. 23 is the focus of Akutagawa Ryûnosuke's literary classic *Kappa*. [Olry]

23^{3}+23^{6}+23^{0}+23^{6}+23^{0}+23^{1}+23^{2}+23^{9}+23^{4}+23^{9}+23^{0}+23^{5}+23^{7} = 3606012949057. [Vosbard]

Let c = 2^a * 3^b with a,b>0 and prime p with p^n, n>1, c-p^k=prime. Is prime=23 the most common? E.g., 48-25=23, 72-49=23, 144-121=23, 192-169=23, 648-625=23, etc. [Bergot]

Square 23 to get 529, square 529 to get 279841. Notice that the last three digits (841) equals 29^2, providing 29 as the next prime after 23. [Bergot]

The fifth Mars rover has a total of 23 cameras. [Wilson]