Zoo of Numbers: Numbers 24 to 69

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What's Special About This Number?

Fun facts on numbers (and their strange properties), both mathematical and cultural, tantalising problems and anecdotes. The number facts in this website are also available as features for print and electronic publishing.

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is smallest number with exactly eight divisors (1, 2, 3, 4, 6, 8, 12, 24), and the largest number with the following property: subtracting one from each of its divisors produces a prime number.

is the number of different ways in which the letters of 'STOP' may be arranged. Only 6 arrangements produce words, though.

Sotp? Oops... ->

= 2¹⁰ - 10³
= 7² - 5² (difference of squares)
= 4!
= 3 + 5 + 7 + 9 (sum of consecutive odd numbers)
= 2⁵ - 2³
= 3³ - 3¹
= (D/C)!/(D/C)

24 = (2 + 4)!

Select any prime number equal or greater than 5, square it, subtract 1 from that, then the result is always divisible by 24. For instance, 37² - 1 = 1368 is divisible by 24. However, this works just as well with any odd number not divisible by 3. For instance, 125² - 1 = 15624 is divisible by 24. In fact, any number of the form p² - q² can be divided by 24, for p and q primes > 3.

The product of 4 consecutive numbers n(n+1)(n+2)(n+3) is divisible by 24.

24 is the largest number of Knights you can place on a chessboard such that none of them can capture any other using the standard chess Knight's moves.

Jogo do Bicho ("the animal game") is an illegal gambling game in Brazil, prohibited by federal law, but very popular throughout the country. In this game, 24 is the number given to the deer (‘veado’ in Portuguese, which sounds like ‘viado’), an animal that has long been pejoratively associated with gay men. Because of that, the number 24 is heavily avoided by Brazilian male athletes, with rare exceptions. Football players, for instance, usually reject this number for their jerseys, and may express dissatisfaction when obligated to wear it.

The Slinky toy (aka "Lazy Spring") is a toy consisting of a 24-meter (18-foot) long coil spring that stretches and can bounce up and down.

There are 24 hours in a day.

The phrase HOURS IN A DAY requires 24 penstrokes!

is a centered octagonal number.
is the smallest square number that can be written as a sum of 2 (consecutive) squares: 5² = 3² + 4²
is an automorphic cubic number: 25³ = 625

= 1 + 3 + 5 + 7 + 9 (sum of the first five odd numbers)
= 3 + 4 + 5 + 6 + 7 (sum of consecutive numbers)

= 1² + 2² + 2² + 4²
= (1² + 2²)²
= 13² - 12²

25² = 7² + 24² = 15² + 25² (Pythagorean triples)

4! + 1 = 25

The powers of 25 (or any other number ending by 25) end with 25: 725² = 525625

When you add the digit 6 to 25 you obtain two squares:
625 = 25², and 256 = 16²

We can readily test for divisibility by 25 by seeing if the last two digits of the number match 25, 50, 75 or 00.

In the Western Occult Tradition, each planet has traditionally been associated with a particular organizations of numbers. One such method of numerological arrangement is the "Magic Square".

Magic Square of Mars has 25 squares:
(magic constant: 65)

25 percent (25%) means one quarter (1/4).
25 percent of gold mass in a material is equivalent to 6 carats.

It seems that 25 and 27 are the only case of a square and a cube separated by 2 (can you prove it?), in fact: 5² + 2 = 3³

The Sun revolves once in 25 days near the Poles and about 30 days near its Equator.

"World Pasta Day" is celebrated on 25th October worldwide.

The 25th Hour is a 2002 Spike Lee urban drama based on David Benioff's novel.

A "pony" is a British slang for £25.

A chiastic saying: "The two greatest highway menaces are drivers under 25 going over 65 and drivers over 65 going under 25".

is the only number sandwiched between a square (5²) and a cube (3³).
is the difference of the second pair of amicable numbers:
1210 - 1184 = 26
is the smallest number which is not a palindrome, but has a square which is: 26² = 676
is the sum of the following series (J. Bernoulli):

∞ ∑ 1		n3
		2n

The 26 letters of the alphabet can make 40,329x10²² different combinations.

Mathematically correct result derived by incorrect calculation method (anomalous cancellation):
26 = 26 = 2
65    65    5

26 = 1 + 7 + 5 + 7 + 6 and 26³ = 17576

= 5 + 8 + 13 (sum of consecutive Fibonacci numbers)
= 1210 - 1184 (difference of pair of amicable numbers)

"God's Number" is the minimum number of moves needed to solve any disordered Rubik's cube. It is so named because God would only need the smallest number of moves to solve a cube! The research has proved that God's number is 26. Nevertheless, theoretical work suggests that God's Number could be in the 'low 20s'.

26 has also a special divine significance in Gematria because it is the sum of the letter-values of the 'Tetragrammaton', the Hebrew biblical name of God ( יהוה Yahweh): Yod + Heh + Wau + Heh = 10 + 5 + 6 + 5 = 26
The diagram above shows the 'Tetragrammaton' written in Phoenician (blue script), Aramaic (red script) and modern Hebrew (black script).

The 26-letter word precipitevolissimevolmente ('as fast as possible'), created in 1677, is one of the longest words in the Italian language. More info on "A collection of word oddities and trivia".

There are 26 letters in the English alphabet.

is the first odd or masculine cube (odd numbers were considered masculine by the Pythagoreans; even numbers feminine... Because they were supposed to be weaker than the odd: in fact odd + even always give odd, and two evens can never produce an odd, while two odds produce an even).

is the number of integer partitions of 15 into distinct parts.

is the smallest 2-digit number in which the sum of digits is equal to the sum of prime factors, e.g.:
27 = 3 x 3 x 3 and 2 + 7 = 3 + 3 + 3 = 9
is the smallest integer which is the sum of 3 squares in two different ways:
1² + 1² + 5² = 27 and 3² + 3² + 3² = 27

is the largest integer which is the sum of the digits of its cube:
27³ = 19,683 and 27 = 1 + 9 + 6 + 8 + 3
The other Integers equal to the sum of the digits of their cubes are: 0, 1, 8, 17, 18, and 26
8³ = 512 and 8 = 5 + 1 + 2
17³ = 4,913 and 17 = 4 + 9 + 1 + 3
18³ = 5,832 and 18 = 5 + 8 + 3 + 2
26³ = 17,576 and 26 = 1 + 7 + 5 + 7 + 6

27³ = 3³ + 18³ + 24³

27 = 3³
27 = 6² - 3² (difference of squares)
27 = 14² - 13² (difference of consecutive squares)
27 = 2 + 3 + 4 + 5 + 6 + 7 (sum of consecutive natural numbers)

27 appears in the resolution of equations of third degree:
D = 4p³ + 27q²

The 27th lucky number is 127.

27 x 198 = 5346 (this equality contains all digits 1 to 9 once)

27! + 1 is prime.

(27/8)^9/4 = (9/4)^27/8

is the magic constant of a prime reciprocal magic square of the multiples of 1/7:

1	4	2	8	5	7
2	8	5	7	1	4
4	2	8	5	7	1
5	7	1	4	2	8
7	1	4	2	8	5
8	5	7	1	4	2

The 27th hexagram of the I-Ching reveals:
THE JUDGMENT
THE CORNERS OF THE MOUTH.
Perseverance brings good fortune.
Pay heed to the providing of nourishment and to what a man seeks to fill his own mouth with.

Twenty-seven is the highest number of the Platonic Lambda series: 1, 2, 3, 4, 9, 8, and 27.

is the number of bones in the human hand.
is the number of "cubies" in a Rubik's cube.

The numbers of atoms in the body of an average male (80 kg, 175 lb) is about 10²⁷.

The musician/humorist 'Weird Al' Yankovic often includes the number 27 somewhere in his songs, videos, album art, and memorabilia. For example, he wears a 27 on the cover of the Running with Scissors album, and 27 photos are included in the photo gallery on the 'Weird Al' Yankovic Live! DVD. This trademark began as several coincidental appearances of the number, but Yankovic began intentionally using it after the original incidents were pointed out to him.

The Twenty-Seven Club: Kurt Cobain, Jimi Hendrix, Janis Joplin and Jim Morrison all died at age 27.

The importance of number 27 in Roman rituals
In 17 AD, to celebrate the Secular Games, the emperor Augustus commissioned a hymn ('Carmen Saeculare') from the poet Horace which was sung by twenty-seven boys and twenty-seven girls.
Every March 16th and 17th, a solemn procession made a circuit of twenty-seven chapels called "Argei" located in various parts of the city of Rome.

The 27-letter word sovramagnificentissimamente ('in a very very very magnificent way'), used by Dante in "De vulgari eloquentia" in the 14th century, was one of the longest words in the Italian language. More info on "A collection of word oddities and trivia".

Buy your favorite Number (27) here.

is an hexagonal number: 28 = 1 + 5 + 9 + 13
is the 2^nd number whose divisors add up to itself (perfect number), e.g.: 1 + 2 + 4 + 7 + 14 = 2²(2³ - 1) [Euclid formula] = 28
is a repfigit number.
is the number of dominoes in standard domino sets.

the string 28 is at position 33 counting from the first digit after the decimal point in Pi.

= 2 + 3 + 5 + 7 + 11 (sum of the first five consecutive primes)
= 1 + 2 + 3 + … + 7 (sum of the first 7 consecutive numbers)

= 10! / 360²
= 8² – 6² (difference of squares)
= 2⁵ – 2² (difference of powers of 2)

28 x 157 = 4396 (this equality contains all digits 1 to 9 once)

A 28-sided polygon is called an icosikaioctagon.

28 digits = 7 palms = 1 royal cubit or "meh" (= 524 mm).

It takes the Moon approximately 28 Earth days to orbit around the Earth. The lunar day is actually a little less than 28 Earth days.

The 28-letter word anticostituzionalissimamente ('in very anticostitutional way') is the longest word in the Italian language. More info on "A collection of word oddities and trivia"

The "Twenty Eight" parrot is an Australian bird, so named because it sounds like the words "28" when it calls.

Recycling: even an old calendar becomes new again every 28 years!

successive primes up to 29 are separated at most by 3 integers (the mean gap being 2).
is the smallest prime of the form 7n + 1.
is the 7^th Lucas number.
is a Perrin number. In mathematics, the Perrin sequence is defined by the recurrence relation: P(0) = 3, P(1) = 0, P(2) = 2;
and P(n) = P(n - 2) + P(n - 3), for n > 2.
is the 3rd number n (following 1 and 5) such that 2n² - 1 is a square, in fact: 2 x 29² - 1 = 41² -- Source: 'Dictionary of Curious and Interesting Numbers' by David Wells
is a Markov number, it appears several times in the Markov triples: (2, 5, 29), (2, 29, 169), (5, 29, 433), (29, 169, 14701)... that define the solutions to the Diophantine equation:
x² + y² + z² = 3xyz
is the exponent of the largest power of 2 whose digits are all distinct: 2²⁹ = 536 870 91
is the last term of the only sexy prime quintuplet:
5, 11, 17, 23, 29
is the smallest multidigit prime whose product of digits of its cube is also a cube:
29³ = 24389 and 2 x 4 x 3 x 8 x 9 = 1728 = 12³

A pizza can be divided into 29 pieces with just 7 straight cuts.

No sum of three 4th powers is divisible by either 5 or 29 unless they all are. -- Euler

= 2² + 5² (sum of 2 squares)

= 2² + 3² + 4² (sum of 3 consecutive squares)

= (6! + (6! + 6)/6)

= (2 x 9) + (2 + 9)

= 2x5 + 3x3 + 5x2 (palindromic expression)

29² = 21² + 20² (Pythagorean triple)

29 + n² is prime for n = 0 to 29-1 (posted by Len Baumann)

3²⁹ - 2²⁹ is prime.

29 can be written as a sum of primes in exactly 3x29 distinct ways.

TWENTY NINE is the only number written with as many strokes as its numerical value.

The probability of being born on 29th February is 1/1,461.
Bob Morris wrote:
Is really 1/1,461 the probability of being born on February 29? If leap years occurred once every four years, it would be, but there are a couple of additional rules for determining if a year is a leap year. If it is divisible by 100, it is not a leap year, but if it is divisible by 400, then it is a leap year (for example, 1900 was NOT a leap year because it is divisible by 100. The year 2000 IS a leap year because it is evenly divisible by 100 and by 400). This means that in any 400 year period, there are 365 x 400, or 146,000, non-leap days. Leap days number 100 (one every four years) - 4 (one every hundred years) + 1 (one every four hundred years), so there are 97 leap days every 400 years.
The probability of being born on February 29 is therefore 97/146,097.

29.783 km/s is the average orbital speed of the Earth around the Sun.

29.53 days is the amount of time for the Moon to complete one orbit around the Earth (synodic month).

The planet Saturn requires over 29 years to orbit the Sun (solar revolution).

is the number of bones in a normal human skull.
is the number of days February has on leap years.

In numerology, 29 is a very unlucky number and indicates uncertainties, treachery and deception from others. It also stands for dangers, unreliable friends, grief and deception from the opposite sex.
In the Tarot, the 29 is represented by the 3 of Wands. It infers spiritual power, introspection, reflection and receptivity.

30

is a pyramidal number - the sum of the first 4 square numbers:
1² + 2² + 3² + 4²
is the largest number such that every smaller coprime to it is prime.

1¹ + 2² + 3³ ... + 28²⁸ + 29²⁹ + 30³⁰ is prime!
- Crespi de Valldaura

= 4 + 5 + 6 + 7 + 8
≈ π³ - π/500

30 + 25 = 55 and 55 x 55 = 3025

n⁵ – n is divisible by 30.

A polygon with 30 sides is called a tricontagon. Both an icosahedron and a dodecahedron have 30 edges.

Used to indicate the end of a wire service story (possibly a corruption of German fertig - "finished, ready" or in journalistic context "end of story"?).

Emile Coué, a physician who formulated the Laws of Suggestion, was known for encouraging his patients to say to themselves 30 times a night before going to sleep: "Everyday in every way, I am getting better and better".

THIRTY in different languages (© G. Sarcone) (Old English thrîtig)

Indo-european | *TRIKOMT(A) | Sanskrit | trimsat | Greek, Attic | τριάκοντα TRIAKONTA | Latin | TRIGINTA |

Italian, Catalan, Provençal and Rumansh trenta; French trente; Spanish treinta; Portuguese trinta; Romanian treizeci; Sardinian trínta.

Old Germanic | *THREJIZTIGUS |

Dutch dertig; German dreißig; Danish tredive or tredve; Norwegian tretti or tredve; Swedish trettio; Icelandic þrjátíu.

Old Slavic | TRIJEDESETI |

Russian тридцать tridtsat’ ; Czech třicet; Slovenian trídeset; Polish trzydzieści.

Chinese 三十 sānshí or 卅 sà (complex form 叁拾).

Ancient Egyptian [mˁbȝ] meobe; Akkadian šalāšā.

Arabic ثلاثون thalathun; Hebrew שלושים shlosheem; Maltese: tletin; Amharic sélasa.

More languages Magyar harminc. Mayan lahuncakal. Nahuatl cēmpōhualmahtlāctli. Suomi kolmekymmentä. Zulu amashumi amatatu.

is the first lucky prime, and the 3rd Mersenne prime (2⁵ - 1).
is very close to ³.

Only 31 numbers cannot be expressed as the sum of distinct squares, there are: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, and 128.

= 2⁵ - 1
= 2² + 3³ (sum of the first two primes raised to themselves)
= 1 + 5 + 5²
= 2⁰ + 2¹ + 2² + 2³ + 2⁴ = 1 + 2 + 4 + 8 + 16
= 3³ + 3 + 3/3

31² x 325 = 312325 (see also 3⁴ x 425 = 34425)

The sum of the first 31 odd primes is a perfect square:
3 + 5 + 7 + 11 + ... + 83 + 87 + 89 = 31²

31
331
3331
33331
333331
3333331
33333331 are all primes
But will there emerge a composite number?
333333331, the 8th term of the sequence, is not prime; it is divisible by 17. What’s more, in this sequence every 15th number is divisible by 31. In fact, any sequence that looks like ab, abb, abbb, abbbb, … (where a and b are digits) contains terms that are periodically divisible by the first 2-digit number ab.
The lesson to be learnt is: you can never assume that statements that hold for small numbers will hold for big numbers!

A billion (1,000,000,000) seconds is 31 years.

Strangely enough, the French expression "être sur son trente-et-un (31)" means 'to be dressed to the nines' in English, and "anar de vint-i-un (21) botó" in Catalan.

Joke: Why do mathematicians (or programmers) always confuse Halloween and Christmas? - Because 31 Oct = 25 Dec ("oct" = October, but also = octal; and "dec" = December, but also = decimal...)

is the lowest number n with exactly 7 solutions to the equation: Phi(x) = n
is the smallest 5^th power (besides 1), and the largest known power with all decimal digits being prime: 2⁵

= 2⁴ + 4² (Leyland number)
= 5 + 7 + 9 + 11
32! - 1 is prime
(32 + 1)! - 1 is also prime

log₂ 32 = 3 + 2
163 / ln 163 ≈ 32 (= 31.9999987...)

The size of a full set of adult teeth in humans, including wisdom teeth.
32° Fahrenheit is the melting point of ice.

is a 12-gonal number.
is the largest positive integer that is not a sum of distinct triangular numbers.
is a palindrome in binary as well as in base 10.

= 0⁰ + 1¹ + 2² + 3³
= 1⁵ + 3³ + 5¹
= 17² - 16² and 17 + 16 = 33
= 7² - 4²
= 1⁵ + 5⁵
= 1! + 2! + 3! + 4! (The sum of the first 4 factorials)

3 x 37 = 111
33 x 3367 = 111,111
333 x 333667 = 111,111,111
3333 x 33336667 = 111,111,111,111
33333 x 3333366667 = 111,111,111,111,111

165033 = 16³ + 50³ + 33³ and
1000165033 = 1000³ + 16³ + 50³ + 33³

33² = 65² - 56² and
33 33² = 65 65² - 56 56²

33! - 1 is a prime number.

1123 is a divisor of 33³ - 1

30 is 33 in base 9.

There are 33 vertebrae in human spine.

33 Bits of Entropy
There are 6.6 billion people in the world. So in order to figure out who you are you only need 33 bits of data about you or any person. More info.

In the ASCII code, 33 represents an exclamation mark.

L33t (sometimes rendered LEET, 1337, or 31337) is an alphabet used primarily on the Internet, which uses various combinations of ASCII characters to replace Latinate letters. It was initially developed as an exclusionary language: a way to encode text so that messages could only be read by the initiated... Example sentence: L337 15 n07 4 c0mm0n 1n73rn37 5p34k 4m0n9 r34l h4x0r. English rendering: 'Leet is not a common internet speak among real hackers'.

33 is a significant number in modern numerology, as it is one of the master numbers along with 11 and 22.

is the word a Romance language speaking patient is usually asked to say when a doctor is listening to his/her lungs with a stethoscope (fr: trente-trois, it: trentatrè, ro: treizeci şi trei, sp: treinta y tres, and por: trinta e três). In fact, the sound TRR...TRR... makes the lungs vibrate making easier to detect a possible respiratory anomaly.

In Spanish, is the smile-to-the-camera word, as is 'cheese' in English. "Diga treinta y tres" ('say thirty-three') is the same as 'say cheese'.

is a Fibonacci and an heptagonal number.
is the smallest number which can be expressed as the sum of two primes in four ways, and also the smallest number to be surrounded by numbers with the same number of divisors (4):
33 = 3 x 11; 34 = 2 x 17; 35 = 5 x 7

34 x 86 = 68 x 43 (palindromic equality)

= 7 + 8 + 9 + 10 (sum of consecutive integers)
= (1 + 2 + 3 + … + 14 + 15 + 16) / 4

Most-perfect magic square with magic constant of 34
Parshvanath Jain Temple in Khajuraho, India, boasts a remarkable most-perfect magic square. It contains numbers 1 to n² and has two extraordinary properties: all 2×2 subsquares sum to 2s, where s = n² + 1, and diagonal pairs separated by n/2 sum to s.

is a tetrahedral number - the sum of the first five triangular numbers: 1 + 3 + 6 +10 + 15

= (10 – 3)(10 – 5) (multiplication of its 10-complement digits)

There are almost 35 hexominoes, the polyominos made from 6 squares.

is the smallest number which is the sum of pairs of distinct odd primes in four ways:
36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19
is the smallest integer (besides 1) being both a square and a triangular number.
is the smallest number to have 9 different factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

2 dice can fall in 36 different manners.

= π / 5
= 3! x 6
= 2² x 3² (perfect square)
= 1³ + 2³ + 3³ = (1 + 2 + 3)²

Repeatedly inserting numbers 36 within the prime number 11 yields other prime numbers: 1361, 136,361, 13,636,361, 1,363,636,361 and 136,363,636,361.

In our Millenium occur 36 palindromic dates in the mm-dd-yyyy format. Palindromic dates are the same when read backward and forward; the first ones are:
10/02/2001, 01/02/2010, 11/02/2011, 02/02/2020, etc.

The 36 officers problem is a puzzle proposed by Swiss mathemacian Leonhard Euler in 1782. The problem asks if it is possible to arrange six regiments consisting of six officers each of different ranks in a 6x6 square so that no rank or regiment will be repeated in any row or column. Such an arrangement would form a Graeco-Latin square. The answer is that no such arrangement is possible...

36 Hidden Holy Persons (Heb. ל״ו צַדִּיקִים, "36 righteous men")
In the Kaballah, the numerical value of the Hebrew letters lamed (L) + vav (V) is 36, and represents the 36 hidden tzaddikim -wise men- whose actions helped to repair the world. The concept of the "Hidden Tzaddikim" goes back to the early days of Chassidism, in the 17th century. According to Jewish folklore these 'hidden saints', called in Yiddish Lamedvavniks, were responsible for the fate of the world and one of them is considered to be the Messiah.

Chinese characters can range in complexity from just one horizontal stroke — yi “one” to 36 strokes, the largest number of strokes in any character actually used: is contained in the word nàng referring to “the sound your voice makes through a stuffed up nose”.

is a hexagonal number because it can be arranged in a pattern of nested hexagons: 37 = 1 + 6 + 12 + 18.
is the maximum number of 5th powers needed to sum to any integer (Hilbert-Waring's problem).

If you ask someone to think of any number from 1 to 50, one people out of three answers 37. Strangely, if you ask someone else to think of any number from 1 to 5, the probability that he/she will answer 3 is one out of three; and if you ask to think of any number from 5 to 10 instead, the probability that he/she will answer 7 is also one out of three! -- G. Sarcone

is a 'psychologically random number' - that is a number which is chosen more often when someone is asked to pick a number (usually this kind of number is odd and doesn't end in 5, because there is a natural psychological bias to think of even numbers and numbers that end in 5 are 'less random'). 37 appears disproportionately in television and movies.

On a calculator, any row, column or diagonal typed forwards and then backwards is divisible by 37. For example:
147741 divided by 37 = 3993
852258 divided by 37 = 23034
753357 divided by 37 = 20361
- Posted by Tim Barss -

A 6-digit number in which the first 3 digits are consecutively-increasing numbers; and the last 3 digits, consecutively-decreasing numbers, is divisible by 37. For instance, the numbers 123987, 234765 and 567543 are all divisible by 37.

= 2² + 2² + 2² + 5²
= 4³ - 3³
37² = 12² + 35² = 19² - 18² (Pythagorean triple)
37³ = 37² + 222² = 107² + 198²

37! + 1 is prime.

1/37 = 0.027027027..., and 1/27 = 0.037037037... (This is related to the fact that 37 x 27=999).

3.703703703703... x 10³⁷ = 3 333 333 333 333³

2³⁷ = 137438953472
37! = 13763753091226345046315979581580902400000000

3² + 7² - 3 x 7 = 37
3 x (3 + 7) + 7 = 37
(3³ + 7³)/(3 + 7) = (4³ - 3³)/(4 - 3) = 37
333/(3 x 3) = 37
33 + 3 + 3/3 = 37

3 x 37 = 111 and 1 + 1 + 1 = 3
6 x 37 = 222 and 2 + 2 + 2 = 6
9 x 37 = 333 and 3 + 3 + 3 = 9
12 x 37 = 444 and 4 + 4 + 4 = 12
15 x 37 = 555 and 5 + 5 + 5 = 15
18 x 37 = 666 and 6 + 6 + 6 = 18
21 x 37 = 777 and 7 + 7 + 7 = 21
24 x 37 = 888 and 8 + 8 + 8 = 24
27 x 37 = 999 and 9 + 9 + 9 = 27

37 is factor/divisor of:
111
111,111
111,111,111
111,111,111,111
etc.

37 uniform counters may be arranged on a flat surface to form any one of three symmetrical patterns shown opposite: hexagon, hexagram and octagon.

37 minutes (and 5 seconds) is about a Golden Section of an hour!

'Normal' human body temperature is approx. 37 degrees Celsius.

The Carthaginian general Hannibal Barca began his trip across the Alps with 37 war elephants: "Annibal relicto in Hispania fratre Hasdrubale, Pyrenaeum transiit; Alpes, adhuc ea parte invias, sibi patefecit. Traditur ad Italiam LXXX milia peditum et XX millia equitum, septem et XXX elephantos adduxisse" (Flavius Eutropius).

Buy your favorite Number (37) here.

is the last Roman numeral when written lexicographically (XXXVIII).
is the largest known even number which can be represented as sum of two distinct primes in only one way: 38 = 31 + 7
is the sum of the squares of the first three primes:
38 = 2² + 3² + 5²

37 and 38 are the first pair of consecutive positive integers not divisible by any of their digits (the second one being 46 and 47...).

= 8 + 9 + 10 + 11 (sum of consecutive numbers)
= 3 + 5 + 6 + 7 + 8 + 9 (unique possible partition with 6 different integers)
= 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 (unique possible partition with 8 different integers)

is the last Roman numeral alphabetically: XXXVIII (38)

The number 38 in Mandarin Chinese (三八 - Sanba - Silly) is an insult referring to someone who acts in a 'girly way'.

In Hong Kong 38 is a lucky number and car registrations like 3838 are much in demand from business people.

is the smallest number which has 3 different partitions into 3 parts which all give the same product when multiplied:
39 = 4 + 15 + 20 and 4 x 15 x 20 = 1200
39 = 5 + 10 + 24 and 5 x 10 x 24 = 1200
39 = 6 + 8 + 25 and 6 x 8 x 25 = 1200

is the smallest positive integer which cannot be formed from the first four primes (used once each), using only the simple operations +, -, x, : and ^.

is the smallest integer with 6 representations as a sum of three distinct primes:
39 = 3 + 5 + 31 = 3 + 7 + 29 = 3 + 13 + 23 = 3 + 17 + 19 = 5 + 11 + 23 = 7 + 13 + 19

= 4 + 5 + 6 + 7 + 8 + 9 = 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 (sum of consecutive integers)

= 3 x 13 = 3 + 5 + 7 + 11 + 13 (sum of a sequence of consecutive primes and product of the first and last primes in that sequence)

Specular multiplication properties:
39 x 31 = 13 x 93
39 x 62 = 26 x 93

= 20² - 19² = 20 + 19
= 8² - 5² (difference of squares)
= 3¹ + 3² + 3³ (sum of the first three powers of 3)

39 x 186 = 7254 (this equality contains all digits 1 to 9 once)

39 weeks = 3 trimesters = length of a human pregnancy

Barbie's measurements if she were life size: 39-23-33.

39 years is the average life expectancy in Mozambique (United Nations, 2005–2010).

"The 39 steps" is the title of the John Buchan novel and subsequent films.

39 is the code for international direct-dialed phone calls to Italy.

40

is a 8-gonal number, and as the sum of the first four pentagonal numbers (1, 5, 12, 22), it is also a pentagonal pyramidal number.

= 2³ x 5
= 6 + 7 + 8 + 9 + 10 (sum of consecutive numbers)
= 3⁰ + 3¹ + 3² + 3³ (sum of powers of 3)
= 2³ + 2³ + 2³ + 2³ + 2³ (sum of cubes)
= (41² - 9²)
= (3⁴ - 1)/2 (number of the form (3ⁿ - 1)/2)

40³ = 80² + 240² = 144² + 208²

40⁴⁰ - 999 is prime.

-40° Celsius = -40° Fahrenheit

is the only number whose letters are in alphabetical order (forty).

It needs approximately 40 grams of petrol to manufacture 1 kg of strawberry yogurt. The so-called ‘environmental’ products are not environmental at all if the distances covered by freight transports are included!

Symbolizes the number of trial and privation, possibly starting when the Babylonians observed the forty-day disappearance of the Pleiades (aka 7 Sisters) which coincided with the rainy season, storms and floods, trial, danger. The Pleiades' return marked the start of the New Year Festival. Other negative associations for forty are:
- 40 years of Hebrew wandering in the desert,
- 40 days and nights of the great flood,
- 40 days of Moses on Sinai,
- 40 days of Elijah's journey,
- 40 days of mourning for Jacob,
- 40 days of Lent, the period of fasting, self-denial, and penitence traditionally observed by Christians in preparation for Easter,
- 40 day period of isolation in the Roman port, which survives in the word quarantine.

But in monotheistic religions, 40 may also be shorthand for 'a long time'. In fact, Jesus spent 40 days fasting in the wilderness; God gave Nineveh 40 days to repent; the great flood lasted 40 days and 40 nights; the Jewish people wandered the desert for 40 years. Imam al-Baqir has said: "The prayers of someone who drinks wine are not accepted for 40 days."

WD-40 is the trademark of a widely used penetrating oil spray. It was developed in 1953 by the chimist, Norm Larsen to eliminate water and prevent corrosion on electrical circuitry. WD-40 stands for 'Water Displacement, 40th attempt'. The name was coined by Norm Larsen itself, while he was attempting to concoct the formula to prevent corrosion by displacing water. Norm's persistence paid off when he perfected the formula on his 40th try.

A typical pregnancy actually lasts 40 weeks.

40 is the number of thieves Ali Baba clashes with in "Arabian Nights".

The French expression "s'en ficher (moquer, foutre) comme de l'an quarante" (literally 'not to care like the year forty') means not to care, not to give a damn. This expression is a distortion of an old French saying "s'en moquer comme de l'Alcoran" (literally 'not to care like the coran', XVIIIth century).

FORTY in different languages (© G. Sarcone) (Old English fêowertig)

Indo-european | *KwETURKOMT(A) | Sanskrit | catvârimsat | Greek, Attic | τετταράκοντα TETTARAKONTA | Latin | QUADRIGINTA |

Italian, Catalan, and Provençal quaranta; French quarante; Spanish cuarenta; Portuguese quarenta; Romanian patruzeci; Rumansh curonta; Sardinian barànta.

Old Germanic | *FITHWORTIGUS |

Dutch veertig; German vierzig; Danish fyrre (fyrretyve archaic); Norwegian førti; Swedish fyrtio; Icelandic fjörutíu.

Old Slavic | CETYRIDESETI |

Russian сорок sorok; Czech čtyřicet; Slovenian štírideset; Polish czterdzieści.

Chinese 四十 sìshí or 卌 xì (complex form 肆拾).

Ancient Egyptian [ḥm.w] Hme; Akkadian erbeā.

Arabic أربعون arba’un; Hebrew ארבעים arba'eem; Maltese: erbgħin; Amharic arba.

More languages Magyar negyven. Mayan cakal. Nahuatl ōmpōhualli. Suomi neljäkymmentä. Zulu amashumi amane.

is a sum of two consecutive squares: 4² + 5²

is the lowest number whose cube is the sum of 3 cube numbers in 2 different ways:
41³ = 2³ + 17³ + 40³ = 6³ + 32³ + 33³ = 68,921

is the smallest non-palindromic prime which on subtracting its reverse gives a perfect cube (41 - 14 = 3³), and is also the smallest number that is not of the form |2^x - 3^y|.

n² + n + 41 and n² - n + 41 produce prime numbers for all integers n, from n=0 to n=40 (see below)

is the first number of a prime number sequence that is 41 numbers long:
41 + 2 = 43
43 + 4 = 47
47 + 6 = 53
53 + 8 = 61
61 + 10 = 71
71 + 12 = 83, and so on... (See the animated gif)
- posted by Ryan Bartling -

is a 'cult number', whose cult status originated with Douglas Adams' use of 42 as 'The Answer' to the ultimate question of life, the universe and everything in his "Hitchhiker's Guide to the Galaxy" series.

A cell holds 42 million protein molecules.

is the area of a scalene triangle having integer sides, with a=7, b=15 and c=20 (Heronian triangle).

is the 5^th Catalan number. Catalan numbers are generated by the formula C_n = [(2n)!/n!(n+1)!]

lies between a twin prime pair: 41, 43

is the he smallest perfect square that is the mean of two cubed twin primes:
42² = (11³ + 13³)/2

= (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³ (sum of three cubes)
= 9 + 10 + 11 + 12 = 13 + 14 + 15 (2 consecutive sums of consecutive integers)
= 0!² + 1!² + 2!² + 3!²
= 2(1 + 2 + 3 + 4 + 5 + 6)
= 2² + 2² + 3² + 5² (sum of four prime squares)
= 6¹ + 6² = 7² - 7¹ (sum and difference of successive powers of the same number)

42 x 138 = 5796 (this equality contains all digits 1 to 9 once)

42 and 27 are arithmetically related:
3(2x7)=42 but 3(2+7)=27; and (42 + 4x2 + 4)/2 = 27

The reciprocal of 42 is the smallest fraction in a sum of 4 fractions to add up to 1: 1/2 + 1/3 + 1/7 + 1/42 = 1

The expression n⁷ - n is divisible by 42.

The expression xy + x + y can never equal 42.

The chance of getting at least one 6 with three dice is 91/216, that is approx. 0.42...

A 'gravity train' - also referred to as the 'Gravity Sled' or 'Gravity Express' - is a theoretical means of transportation intended to go between two points on the surface of a planet, following a straight tunnel that goes directly from one point to the other through the interior of the planet. All possible gravity trains on a given planet would take exactly the same amount of time to complete a journey (the length of a trip depends only on the density of the planet and the gravitational constant). For Earth, this time would approximatively equal 42 minutes.

Buy your favorite Number (42) here.

is the smallest non-palindromic prime which on subtracting its reverse gives a perfect square: 43 - 34 = 3²

is the number of sided 7-iamonds (heptagonal number).

is the minimal sum on each face of Honaker's Magic Die - a die with pips (dots) numbered with distinct integers to sum up to the same sum on each face.

is the smallest number, which draws up as sum of two, three, four, or even five different primes:
= 2 + 41
= 3 + 17 + 23
= 11 + 13 + 19
= 2 + 5 + 7 + 29
= 2 + 11 + 13 + 17
= 3 + 5 + 7 + 11 + 17 ...

= 4² + 3³
= 22² - 21² (difference of two consecutive squares)
= 2³ + 2³ + 3³
= 6⁰ + 6¹ + 6² (sum of powers of 6)

43⁷ = 271818611107 = (2+7+1+8+1+8+6+1+1+1+0+7)⁷

43^7/23 = 3.141539853... (a real accurate approximation of )

43 + 4³ is prime.

is the smallest prime that can grow 5 times by the right:
43 is prime,
439 is prime,
4391 is prime,
43913 is prime,
439133 is prime,
4391339 is prime.
...and the smallest prime that can grow 14 times by the left:
443 is prime,
8443 is prime,
18443 is prime,
918443 is prime,
3918443 is prime,
53918443 is prime,
653918443 is prime,
7653918443 is prime,
27653918443 is prime,
427653918443 is prime,
3427653918443 is prime,
33427653918443 is prime,
933427653918443 is prime,
6933427653918443 is prime.

The 43th term of the Fibonacci's sequence is 433494437 and is prime!

43 is the 4th term of the Sylvester's sequence:
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, ...
This sequence is generated by taking the product of the first n terms and adding 1 (for instance: 2 x 3 x 7 + 1 = 43). Because of the way each term is calculated, no two terms in the sequence have a common factor. Then, no two terms have a common prime factor, and thus there are an infinite number of primes!

Licor 43, also known as "Cuarenta Y Tres" (meaning 'forty-three' in Spanish), is a liquor made in Spain for over a thousand years dating back to the time of the Carthaginians (?). It is a bright yellow color liquor that has a sweet citrus-vanilla flavor. 'Licor 43' derives its name from the fact that it is a mixture of 43 basic elements, principally fruit and fragant herbs.

The number 43 symbolically represents the principle of deliverance or change.

43th I-Ching hexagram: |||||¦ is named 夬, guài ('parting') and means resolution, break-through.

Londinium (now London) was founded by the Romans in AD 43.

It requires 43 muscles to frown.

"This morning, I wrote a cheque for € 43-million. That's what I paid for the last fiscal year" - Silvio Berlusconi, former Italian prime minister. June 15, 2007.

is the number of derangements of 5 objects, or said in another manner, the number of different ways in which 5 people with 5 hats can each take the wrong hat...

= 12² - 10²
= !5 = 5! (1-1/1!+1/2!-1/3!+1/4!-1/5!)

In the following equations x=44:
x² + y² = a²
y² + z² = b²
z² + x² = c²

is a Kaprekar number (see further below).
is a triangular number:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

The sum of the first 9 digits placed in increasing or decreasing order, 123456789 and 987654321, gives obviously 45. And the difference between these two numbers gives a result of nine digits, 864197532, of which the sum of each one gives also 45.

= (10+2) + (10-2) + (10x2) + (10/2)
= 3² + 6²
= 6! / 2⁴ = 6! / 4²

45² = 2025
20 + 25 = 45
45³ = 91125
9 + 11 + 25 = 45
45⁴ = 4100625
4 + 10 + 06 + 25 = 45

45³ = 54² + 297² = 135² + 270² (sums of 2 squares)

is a graphical self-reference number in French:
QU4R4NTE+C1NQ

is the number of different possible arrangements of 9 non-attacking Queens on a 9 by 9 chessboard (that means no two Queens should be orthogonally or diagonally aligned on the chessboard).

Human beings have 46 chromosomes in every cell of their body.

is a prime and a Lucas number.
represents the highest number of cubes that cannot tile a cube.
is a repfigit number.
There are 47 occurrences of '47' in the first thousand prime numbers.

= 2⁵ + 5² - 2 x 5
is self-conscious (it refers to itself):
4 + 7 = 11, 7 + 11 = 18, 11 + 18 = 29, and 18 + 29 = 47
47 + 2 = 49 and 47 x 2 = 94

= 24² - 23² = 24 + 23
4³ + 7³ = 407
666⁴⁷ has a digit sum of 666.

9999999999999999999999999999999999999999999999 = 47
 212765957446808510638297872340425531914893617

is a cult number; some people claim that the number forty-seven occurs in nature with noticeably higher frequency than other natural numbers.
is the number of samurai involved in the famous historical Ako vendetta; sometimes referred to as the '47 Ronin'.
The number 47 mysteriously appears in many episodes of Star Trek: 'The Next Generation', 'Voyager', and 'Deep Space Nine'.

A person uses approximately fifty-seven sheets of toilet paper each day.

Buy your favorite Number (47) here.

is the smallest number with 10 proper divisors. Actually, 48 = 2⁴ x 3 and thus is divisible by 2, 3, 4, 6, 8, 12, 16, and 24.
is also the smallest even number that can be expressed as a sum of two primes in 5 different ways:
5 + 43, 7 + 41, 11 + 37, 17 + 31, and 19 + 29
is the double factorial of 6, four dozens, a 17-gonal number, and a Harshad number.
is an arc-cotangent irreducible number, that is a positive integer n for which the greatest prime factor of n² + 1 meets or exceeds 2n: 48² + 1 = 5 x 461 and 461 > 2 x 48
The product of all 10 divisors of 48 equals 48⁴.

= 13² - 11² = 8² - 4² = 7² - 1²
= (4³ + 8³) / (4 + 8)
= 3 + 5 + 7 + 9 + 11 + 13 (sum of consecutive odd numbers)

48 x 159 = 7632 (this equality contains all digits 1 to 9 once)

48² = 2² + 5² + 8² + 11² + 14² +17² + 20² + 23² + 26² (sum of squares of 9 terms of an arithmetic progression of common difference 3)

is the number of hours in two days.

In Italian "succede un quarantotto" (it's going to turn into a forty-eight) means all hell breaks - or will break - loose. While "mandare qualcuno a carte quarantotto" (to send someone to cards forty-eight), means to send someone to hell.

Siddhartha Gautama, the founder of Buddhism, sat under a bo tree for 48 days attempting to figure life out. Buddhism was the result.

With 48 matchsticks you can make a triangle in 48 different ways. - Source: Richard Phillips, "Numbers, facts, figures & fiction"

is the smallest number with the property that it and its bordering numbers are squareful:
48 = 4² x 3, 49 = 7² x 1, 50 = 5² x 2

is a perfect square: it is the smallest number which is the concatenation of two prime squares (2² and 3²) creating another prime square.

is a 'lucky numbers'. Stanislaw Ulam, a polish mathematician, devised a sequence of numbers called lucky numbers which are generated by a simple algorithm called 'sieve' similar to the 'Sieve of Eratosthenes'. The lucky numbers are: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, etc.

Its cube ends in the same digits: 49³ = 117,649

2 fourth powers sharing exactly the same digits:
49⁴ = 5,764,801 and 32⁴ = 1,048,576

49 = (4 x 9) + (4 + 9)
This also works for 19, 29, 39, 59, 69, 79, 89 and 99, for instance: 79 = 7 x 9 + 7 + 9

49 = 47+2 and, reversely 94 = 47x2
499 = 497+2 and, reversely 994 = 497x2
4999 = 4997+2 and, reversely 9994 = 4997x2 ...

is the smallest number with exactly 8 representations as a sum of three distinct primes:
49 = 3 + 5 + 41 = 3 + 17 + 29 = 5 + 7 + 37 = 5 + 13 + 31 =
= 7 + 11 + 31 = 7 + 13 + 29 = 7 + 19 + 23 = 13 + 17 + 19 (sum of consecutive primes)

is a repeating decimal with a period of 42:
1⁄49 = 0.0204081632 6530612244 8979591836 7346938775 51 (42 repeating digits)
The first 10 decimals of 1/49 represent successive powers of 2: 0,0204081632...

= 1 + 3 + 5 + 7 + 9 + 11 + 13 (sum of the first seven odd numbers)
= 7² = 25² - 24² = 25 + 24
= (11 - 4)^{(11 - 9)}
= 3² + 5² + 3 x 5
= 3² + 10 x 2²
= 5² + 6 x 2²

49 = 4 + 9

Illicit semplification:
49 = 49 = 4
98    98    8

An algebraic expression that is always divisible by 49:
2³ⁿ⁺³ - 7n - 8

XLIX or IL?
Rules regarding Roman numerals often state that L cannot be preceded by I or V, only by X. Thus, one should represent the number 'forty-nine' as XLIX, not as the 'shortcut' IL.

During the Gold-Rush, the gold-seekers that came to California were called 'forty-niners' because most of them came in 1849.

Lasca is an unusual checkers game played on a 7x7 board having 49 squares.

The "Nagari" or "Devanagari" (also called 'writing of the gods') is a writing system employed for the traditional Sanskrit language and has 49 elementary signs.

A typical artificial strawberry flavor, like the kind found in a Burger King strawberry milk shake, contains the following 49 ingredients:
amyl acetate, amyl butyrate, amyl valerate, anethol, anisyl formate, benzyl acetate, benzyl isobutyrate, butyric acid, cinnamyl isobutyrate, cinnamyl valerate, cognac essential oil, diacetyl, dipropyl ketone, ethyl acetate, ethyl amyl ketone, ethyl butyrate, ethyl cinnamate, ethyl heptanoate, ethyl heptylate, ethyl lactate, ethyl methylphenylglycidate, ethyl nitrate, ethyl propionate, ethyl valerate, heliotropin, hydroxyphenyl-2-butanone (10 percent solution in alcohol), a-ionone, isobutyl anthranilate, isobutyl butyrate, lemon essential oil, maltol, 4-methylacetophenone, methyl anthranilate, methyl benzoate, methyl cinnamate, methyl heptine carbonate, methyl naphthyl ketone, methyl salicylate, mint essential oil, neroli essential oil, nerolin, neryl isobutyrate, orris butter, phenethyl alcohol, rose, rum ether, gamma-undecalactone, vanillin, and solvent.

Buy your favorite Number (49) here.

is the smallest number that can be written as the sum of 2 squares in 2 distinct ways: 1² + 7² and 5² + 5²
is also the sum of 3 squares: 3² + 4² + 5²
and the product of complex numbers: (7 + i) (7 - i) =
= 49 + 7i - 7i - i² = 49 - (-1) = 50

= 2(1 + 3 + 5 + 7 + 9)
= 8 + 9 + 10 + 11 + 12 (sum of consecutive integers)
= 3 + 11 + 13 + 23 = 2 + 5 + 7 + 17 + 19 (sum of two different sets of consecutive prime numbers)

50 = 19 + 43/2 + 76/8
50 = 23 + 84/6 + 91/7
50 = 28 + 36/4 + 91/7
50 = 37 + 24/6 + 81/9
50 = 39 + 21/7 + 64/8
(the 5 equalities above are pandigital that is all digits from 0 to 9 were used once)

The 50-50/90 rule: Anytime you have a 50-50 chance of getting something right, there's a 90% probability you'll get it wrong.

FIFTY in different languages (© G. Sarcone) (Old English fîftig)

Indo-european | *PENKwEKOMT(A) | Sanskrit | pancâsat | Greek, Attic | πεντήκοντα PENTEKONTA | Latin | QUINQUAGINTA |

Italian, Catalan, and Provençal cinquanta; French cinquante; Spanish cincuenta; Portuguese cinqüenta; Romanian cincizeci; Rumansh tschunconta; Sardinian chimbànta.

Old Germanic | *FIMFITIGUS |

Dutch vijftig; German fünfzig; Danish halvtreds (halvtredsindstyve archaic = 2 1⁄2 x 20); Norwegian femti; Swedish femtio; Icelandic fimmtíu.

Old Slavic | PEnTIDESETJ |

Russian пятьдесят pyat'desyat; Czech padesát; Slovenian pétdeset; Polish pięćdziesiąt.

Chinese 五十 wǔshí (complex form 伍拾).

Ancient Egyptian [dy.w] diywe; Akkadian khamšā.

Arabic خمسون khamsun; Hebrew חמישים khameesheem; Maltese: ħamsin; Amharic amsa.

More languages Magyar ötven. Mayan lahunyoxkal (10 to 60). Nahuatl ōmpōhualmahtlāctli (2 x 20 and 10). Suomi viisikymmentä. Zulu amashumi amahlanu.

is a pentagonal number as well as a centered pentagonal number (one of the few numbers to be both) and an 18-gonal number.

is the smallest number which can be written with all the digits from 1 to 5 (without repetition) as a sum of prime numbers:
2 + 3 + 5 + 41

A scalene triangle having sides 51, 52 and 53 (consecutive integers) has an area that is also an integer (1170 units²). Such triangles whose sidelengths and area are all integers are called Heronian triangles.

There are 51 and 3/7 degrees in the angle of the Egyptian pyramids, which is 1/7 the arc length of a circle.

= (10 + 7)(10 - 7)
= 6 + 7 + 8 + 9 + 10 + 11 (sum of consecutive integers)
= 26² - 25² = 26 + 25
= 2³ + 2³ + 2³ + 3³ (sum of cubes)

is the 'Anti-Devil' number.
According to Abellio, 51 symbolizes the "Universal Time".

"51% is not a mandate!" is an anti-Bush slogan.

The grain used in bourbon whisky cannot be less than 51% maize, with the remainder being made up of malt and rye.

451 is, in Fahrenheit, the ignition temperature of paper.

Pastis 51 is one of favorite anise flavoured drinks of the French people. 'Pastis 51' derives its name from first being produced in 1951.

51 is also the name of a Brazil's leading brand of cachaça (kind of rhum).

Don’t be baffled if you see the number 51 cropping up in Chinese website names, since 51 sounds like 'without trouble' or 'carefree' in Chinese.

Joke: I'm LIVID! As I can't remember what 51, 6 and 500 are in Roman numerals...

Buy your favorite Number (51) here.

is the 5^th Bell number (Bell numbers is the number of distinct possible ways of putting n distinguishable balls into one or more indistinguishable boxes).
is a decagonal number (4 x 4² - 3 x 4).
is a centered 17-gonal number.
is the 3^rd untouchable number (an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer, including the untouchable number itself).

= 5² + 3³
= 23 + 29 (sum of consecutive primes)
= 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 (sum of the 3^rd through 10^th integers)
= 2 + 3 + 5 + 8 + 13 + 21 (sum of the 3^rd through 8^th Fibonacci numbers)

'Genjiko' is the name of a popular Japanese design whereby five vertical bars are either connected or non-connected at the top resulting in 52 design combinations (see picture opposite).
Its name originates from a traditional Japanese game in which 5 incense sticks are randomly taken out from a bag containing 25 incense sticks (5 bags x 5 different types of incenses) and are lit one after another. The player draws 5 vertical lines on a piece of paper and adds a horizontal bar on the top section of the vertical lines each time they smell the same scent.

The microcentury is 52 minutes and 35.76 seconds and was introduced by Enrico Fermi as the "standard" duration of a lecture period. It's equal to exactly 3155.76 seconds, as one millionth of the scientific Julian century, which is defined to be equal to 36525 days of 86400 seconds each.

is the number of weeks in a year.
is the number of cards in a standard pack without jockers
is the number of white keys on the piano (notes in C major scale).
is a significant number in the Maya calendar.

Patolli (or Patole, Spanish form) is an Aztec board game utilizing 52 squares arranged in a cross form. Its name came from the Aztec word for bean - patolli, meaning fava or kidney bean.

52^nd Verse of Buddha's Dhammapada: On Flowers
Like a beautiful flower, brightly coloured and scented,
even so useful is the well-uttered speech of one who acts accordingly.
-- Buddha (240 B.C.)

is the smallest prime which is not the difference between powers of 2 and 3.
is the only two-digit number that is reversed in hexadecimal (base 16): 35₁₆

xy + x + y never yelds 53.

the reciprocal of 53 (1/53) has 13 digits that repeat:
0.01886792452830188679245283....

is the sum of five consecutive primes: 5 + 7 + 11 + 13 + 17

the sum of the first 53 integers is 1431, which is divisible by 53, a property shared by many other numbers.

the sum of the first 53 primes is 5830, which is divisible by 53, a property shared by few other numbers.

the sum of all consecutive prime numbers from 7 to 53 is the product of 7 x 53:
7 x 53 = 7+11+13+17+19+23+29+31+37+41+43+47+53 = 371

53 = 2² + 7² = 1² + 4² + 6² (sum of squares)
53² = 28² + 45²
53⁵ = 418,195,493 = (4+18+1+9+5+4+9+3)⁵
53⁷ = 1,174,711,139,837 = (1+1+7+4+7+1+1+1+3+9+8+3+7)⁷

2⁵³ = 9007199254740992 is the smallest power of 2 containing two successive zeroes.

the chance that no pair of 53 people in a room have the same birthday is approximately 1/53.

is the code for international direct dial phone calls to Cuba.
is the number of countries in Africa.

is a 19-gonal number.
is the smallest number that can be written as the sum of 3 squares in 3 different ways:
7² + 2² + 1² = 6² + 3² + 3² = 5² + 5² + 2²

= 3³ + 3³ = = 3⁴ - 3³
= (3³ + 3² + 3¹ + 3⁰) + (3³ - 3² - 3¹ - 3⁰)

is a heptagonal number, and a centered nonagonal number.
is also the largest triangular number in the Fibonacci sequence:
= 1 + 2 + 3 + … + 8 + 9 + 10
is a palindromic number, and a square pyramidal number:
= 1² + 2² + 3² + 4² + 5²

= 3 + 6 + 10 + 15 + 21 (sum of consecutive triangular numbers)

= 28² - 27² = 28 + 27
= 8² - 3²

= 2⁰ + 2¹ + 2² + 2³ + 3⁰ + 3¹ + 3² + 3³
= 10² - 9² + 8² - 7² + 6² - 5² + 4² - 3² + 2² - 1²

5³ + 5³ = 250 and 2³ + 5³ + 0³ = 133
and 1³ + 3³ + 3³ = 55

55² = 44² + 33²
55² = 3025 and 30 + 25 = 55

55⁷ = 1,522,435,234,375 --> 1 + 5 + 2 + … + 7 + 5 = 46
46⁷ = 435,817,657,216 --> 4 + 3 + 5 + … + 1 + 6 = 55

The reciprocal of 55 has a curious relationship to the Fibonacci sequence: 1/55 = F(n) x 8^-(n+1) = 0.0181818182

Every integer greater than 55 is the sum of distinct primes of the form 4n + 3.

55 multiplied by any of the odd numbers comprised between 90 and 110 produces a palindrome numeral.

1 to 10 the sum is 55
1 to 100 the sum is 5050
1 to 1000 the sum is 500500
1 to 10000 the sum is 50005000
1 to 100000 the sum is 5000050000
1 to 1000000 the sum is 500000500000

A classic coin puzzle involving number 55
There are 10 hats containing each 17 coins.
9 hats contain "identical real" coins - weighing 10 grams each.
But 1 hat contains 17 "identical" fake coins. Fake coins are 1 gram off (so all 17 coins are either heavier or lighter).
All 170 coins are identical in every other characteristics. Your mission is to locate the "fake" coins by using an electronic balance once!
Solution
Take just 1 coin from the hat #1; 2 from hat #2; 3 from hat #3, and so on... In order to get:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 coins.
Then measure the mass of the 55 coins. Obviously, if ALL the coins were real the mass would be: 10 x 55 = 550 grams.
Now, if the actual mass is off by 1 gram, then the fake coins are in hat #1. If by 2 grams, hat #2; by 3 grams, hat #3... In fact, the difference in grams between 550 and the actual weighing gives the rank of the hat containing the fake coins.

The "7-38-55 rule" states that ommunication is 7% verbal, 38% tone of voice, and 55% body language.

The 55th wedding anniversary is traditionally named "Emerald Anniversary".

is a tetrahedral number:
= 1 + 3 + 6 + 10 + 15 + 21 (sum of the first 6 triangular numbers)

= 3 + 5 + 7 + 11 + 13 + 17 (sum of consecutive primes)

56² = 65² - 33²

is the maximal number of regions into which 10 lines divide a plane.

is the number of reduced 5 x 5 Latin squares.

is the 12th term of the partition numbers, which count the number of ways of placing N indistinguishable items into N indistinguishable urns (but with no empty urns).

56.9612484322608203... is the value of x, for which x^x=10¹⁰⁰. It is therefore the biggest number you can raise to its own power on most scientific calculators.

(111+1)/(1+1) = 56
(222+2)/(2+2) = 56
(333+3)/(3+3) = 56
...
(999+9)/(9+9) = 56

The '56 Aubrey Holes' are one of the oldest features of Stonehenge. They are 56 holes arranged in a circle, but their purpose is still unknown.

= 3 x 19
= 111 in base 7 (that is: 57 = 7² + 7¹ + 7⁰)
= 7 + 8 + 9 + 10 + 11 + 12
= 1² + 2² + 4² + 6² (sum of squares)
= 29² - 28² = 29 + 28
is a 20-gonal number.
can be expressed as a sum x^y + y^x, where x and y are integers greater than 1: 57 = 2⁵ + 5². Such numbers are called 'Leyland numbers'.

The number of ways that 57 can be partitioned into whole number parts no larger than 2 is: 29.
The number of ways that 57 can be partitioned into whole number parts no larger than 3 is: 300.

57 degrees and 18 minutes is the number of degrees in 1 radian.

The 57th day of the year = February 26

57 is the number of sauce varieties H. Heinz lied about to promote his famous ketchup (when he began his business, he actually had many fewer than 57 sauces or food products!). The Heinz people are still quite attached to the number 57: the phone number at corporate headquarters in Pittsburgh is 237-5757, and the address is P.O. Box 57. "Heinz 57" is also used as a joke slang term for mixed-breed dogs, as well as people who are multiracial, or most commonly, people who are of different national heritages.

The most complex Chinese character still in use is the sinograph 'biáng', with 57 strokes! Which refers to Biang biang noodles, a type of noodle from China's Shaanxi province.

is the sum of the first seven prime numbers:
2 + 3 + 5 + 7 + 11 + 13 + 17

= 13 + 14 + 15 + 16 (sum of consecutive numbers)
= 2² + 2² + 5² + 5²
= 3² + 7²

is a 'hoax number', that is a composite number the sum of whose digits is the sum of the digits of its distinct prime factors. In this case:
5 + 8 = 2 + (2 + 9) -- prime factors of 58 being 2 and 29
58 reversed (85) is also a hoax number, in fact:
8 + 5 = 5 + (1 + 7) -- prime factors of 85 being 5 and 17

58° Celsius is the highest air temperature ever recorded on earth (recorded in Lybia, 1922).

The Welsh name for a village on Anglesey Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch (also spelt Llanfair Pwllgwyngyll and commonly known as Llanfair PG) has 58 letters.

is the smallest number which when divided by 2, 3, 4, 5, and 6 leaves remainders of 1, 2, 3, 4 and 5, respectively.
is a "safe prime". That is, a prime of the form 2p + 1, where p is also a prime: 59 = 2 x 29 + 1

59 and 61 are twin prime pair.

59 divides 616161... 61 (repeated 29 x n times, n integer), but 61 divides 595959... 59 (repeated 30 x n times, n integer).

= 17 + 19 + 23 (sum of consecutive primes)
= 30² - 29² = 30 + 29
= 2³ + 2³ + 2³ + 2³ + 3³ (sum of cubes)

59⁴ = 133⁴ + 134⁴ - 158⁴ (Euler argued that there were solutions to the equation a⁴+ b⁴ = c⁴+ d⁴, where a < c < d < b are positive integers. The first solution is in fact a = 59)

59 inches = 1 yard + 1 foot + 11 inches

is the number of times the minute and second hands on a clock are aligned in one hour.

1 day on planet Mercury lasts about 59 Earth days.
Only 59% of the Moon's surface is directly visible from the Earth.

is an abundant number.
is adjacent to 2 prime numbers: 59 and 61.
is also an heptagonal pyramidal number and a 21-gonal number (icosihenagonal number of the form (19n² - 17n)/2).
is the smallest number divisible by 1 through 6 and the smallest number with 12 different divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60).
is the smallest number which is the sum of two odd primes in 6 different ways: 60 = 29 + 31 (twin primes) = 23 + 37 = 19 + 41 = 17 + 43 = 13 + 47 = 7 + 53
is the product of Pythagorean triples: 3 x 4 x 5

= 5 + 7 + 9 + 11 + 13 + 15 (sum of consecutive odd numbers)
= 8 + 10 + 12 + 14 + 16 (sum of consecutive even numbers)
= 29 + 31 = 11 + 13 + 17 + 19 (sum of consecutive primes)
= 8² - 2² = 16² - 14² (difference of squares)
= 2⁶ - 2² (difference of powers with same base)
= 4 x 4 x 4 – 4 = 44 + 4 x 4 = 4⁴/4 - 4 (four 4's diversion)

Base 60 was the number system used by the Mesopotamians and is the reason we have 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle, etc. - Posted by Tim Barss

60 is the last term of a sequence of increasing integers which can be spelled in 5 letters: 3, 7, 8, 20 (score), 40, 50, 60 - G. Sarcone

Each internal angle in an equilateral triangle = 60 degrees.
29 February (the leap day) is the 60th day of the year.
60 is the international telephone dialing code for Malaysia.
‘Diamond wedding’ anniversary celebrates 60 years of marriage.
60 is the atomic number of Neodymium (Nd) = 60 (60 protons and 60 electrons).
The cycle of 60 in the Chinese zodiac (六十花甲; liùshí huājiǎ) is formed by the 5 Elements and 12 terrestrial Branches (Animals).

In our Millenium occur 60 palindromic dates in the dd-mm-yyyy format. Palindromic dates are the same when read backward and forward; the first ones are:
10/02/2001, 20/02/2002, 01/02/2010, 11/02/2011, etc.

Did the Ancient Greeks tell jokes? Yes, they did! Sources tell of joke-groups; who met to trade and test their wit, like the Group of Sixty who met in the temple of Heracles in Athens in the 4th century BC, and whom even Philip of Macedon paid to send him a collection of their best jokes (cf. Athenaeus, "Wise Men At Dinner", 614d).

Fun with degrees
What drink does this formula represent:
60(3 degree arc + 6 degree arc + 3 degree arc) + A
90(2 degree arc + 4 degree arc + 1 degree lines) + A
Answer

SIXTY in different languages (© G. Sarcone) (Old English siextig)

Indo-european | *S(w)EKSKOMT(A) | Sanskrit | s'as'tih | Greek, Attic | ἑξήκοντα HEXÊKONTA | Latin | SEXAGINTA |

Italian sessanta; French soixante; Spanish sesenta; Portuguese sessenta; Catalan seixanta; Provençal seissanta; Romanian şaizeci; Rumansh sissonta; Sardinian sessànta.

Old Germanic | *SEKSTIGUS |

Dutch zestig; German sechzig; Danish tres (tresindstyve archaic = 3 x 20); Norwegian seksti; Swedish sextio; Icelandic sextíu.

Old Slavic | ShESTIDESETJ |

Russian шестьдесят shest’desyat; Czech šedesát; Slovenian šéstdeset; Polish sześćdziesiąt.

Chinese 六十 liùshí (complex form 陆拾).

Ancient Egyptian [sỉsy.w or ỉswy.w (?)] se; Akkadian šeššā.

Arabic ستون sittun; Hebrew שישים sheesheem; Maltese: sittin; Amharic selsa.

More languages Magyar hatvan. Mayan oxkal ( 3 x 20). Nahuatl yēmpōhualli (3 x 20). Suomi kuusikymmentä. Zulu amashumi ayisitupa.

is a cuban prime of the form p = (x³ - y³)/(x - y),
where x = y + 1 and y > 0.
is a Pillai prime, since 8! + 1 is divisible by 61, but 61 ≠ 1 + 8n, where n = any integer.
is the smallest prime whose reversal (16) is a square.
is the 6^th Euler number.
is a repfigit number.

= 5³ - 4³
= 5² + 6²
= 31² - 30² = 31 + 30

61² = 11² + 60² (Pythagorean triple)

is the smallest prime such that its square can be expressed as the sum of 1, 2, 3, and 4 distinct squares:
61² = 60² + 11² = 52² + 24² + 21² = 56² + 22² + 10² + 1²

59 and 61 are twin prime pair.

The 61st Fibonacci number (2,504,730,781,961) is the least Fibonacci number that contains all the digits from 0 to 9.

61 items can be arranged in a regular pattern to make a hexagon with 5 items per side.

1 liter ≈ 61 inches³

“61*” is also the name of TV show/series or movie.

is the smallest number that can be written as the sum of of 3 distinct squares in 2 distinct ways:
= 1² + 5² + 6²
= 2² + 3² + 7²

= 2³ + 3³ + 3³ (sum of cubes)

is the smallest number with 7 divisors, and the smallest power of 2 with no prime neighbors.
is the square of 8, the cube of 4, and the 6th power of 2:
2⁶ = 4³ = 8² = 64
is the difference of the first pair of amicable numbers:
284 - 220 = 64
is the number of ways of distributing 7 identical objects into one or more different containers.

= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 (sum of consecutive odd numbers)
= 17² - 15²
= 10² - 6²
= 2⁷ - 2⁶
= (2² + 2²)(2² + 2²) = 4
= (2 +/- 2i3)³ = (1 +/- i3)⁶ (power of complex numbers)

= 7 x 3² + 1² (Any number which is both square and cube is of the form 7k or 7k + 1)

64 x 23 = 32 x 46 (mirror number pattern)

How to write 64 with just two 4's: ((4))^4!

The number 2⁶⁴ + 1 (≈ 1.84467441 x 10¹⁹) can be factorized!

The formula xy + (x + y) never produces 64.

is the number of hexagrams in the I Ching.

There are 64 squares on a checker/chess board.
In China, "64 incident" refers to the Tiananmen Square protests which happened on 4 June 1989.

appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).
is the lowest integer that becomes square if its reverse is either added to or subtracted from it.
is the 'magic constant' of a 5x5 magic square.
The protein in egg whites coagulate at 65 degrees C (150 degrees F).
In the ASCII code used by computers, 65 represents a capital letter A.

= 1⁵ + 2⁴ + 3³ + 4² + 5¹ - posted by Jim O'Donohoe
= (63² + 16²)
= (56² + 33²)
= 1² + 8²
= 4² + 7²
= 33² - 32²
= 2 + 3 + 4 + ... + 9 + 10 + 11
65 + 56 = 121 = 11²
65 – 56 = 9 = 3²
65² – 56² = 33² (Pythagorean triple)

Magic Circle
If you add up the numbers in each spoke OR in each ring of the circle below, you will always obtain 65.

is a triangular palindromic number and the number of 8-iamonds.

= 1 + 2 + 3 + … + 10 + 11
= 2² + 2² + 3² + 7² (sum of square numbers)

Is the largest number <100 that contains no letter 'e' (sixty-six). The next number with no embedded 'e' is number 2000 (two thousand). Such numbers are called 'eban numbers' (i.e. 'e' is banned).

There are 66 books in the Bible.

With just 12 cuts it is possible to divide a pizza into 67 tiny pieces.

1/67 written as decimal is 0.0149253731... and continues as a repeating sequence of 33 digits.

is the smallest composite number that becomes prime by turning it upside down.

= 1 x 2² x 17
= 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
= 14 + 16 + 18 + 20

68.8 = 8 · 8.6 (related to vampire numbers)

The last day of November 46 B.C. was the longest day in history, it lasted 68 days!

In May '68 a general strike broke out across France. It quickly began to reach near-revolutionary proportions before being discouraged by the Stalinist oriented French Communist Party, and finally suppressed by the government, which accused the Communists of plotting against the Republic.

is the fifth number that stays the same when written upside down (such numbers are called strobogrammatic), the complete list of invertable numbers starts: 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119...
As well as 6 and 9 being the same digit rotated 180 degrees, they are also binary complements:
6 = 0110
9 = 1001

69 ≈ 163( - e) [...within 0.0005%]

has the property that n² and n³ all together contain each digit once: 69² = 4,761 and 69³ = 328,509

is often the largest factorial that can be calculated by a pocket calculator: 69!≈1.71x10⁹⁸ (less than 10¹⁰⁰)

69³² = 6.96969696 · 10⁵⁸

69² =    4761
69³ = 328509 (the two answers include all digits once)

The title of the novel 69 by Ryu Murakami.
This number symbolizes also an erotic position... No need to describe it!

is the Zodiac sign of the Cancer.

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Encyclopedia of Numbers © G. Sarcone, Archimedes Lab, Genoa, Italy