Zoo of Numbers: Numbers 24 to 69

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.

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

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 divisible 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.

There are 24 hours in a day.

is a centered octagonal number.
is the smallest square number that can be written as a sum of 2 squares: 5² = 4² + 3²

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

= (1² + 2²)²
= 13² - 12²

25² = 7² + 24² (Pythagorean triple)

4! + 1 = 25

25% = 1/4

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

It seems that 25 and 27 are the only case of a square and a cube separated by 2 (can you prove it?).

"The two greatest highway menaces are drivers under 25 going over 65 and drivers over 65 going under 25".

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

is the only number sandwiched between a square (5²) and a cube (3³).
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.

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 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 27th lucky number is 127.

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

= 3³ = 5² + 2
= 6² - 3² (difference of squares)
= 14² - 13² (difference of consecutive squares)
= 2 + 3 + 4 + 5 + 6 + 7 (sum of consecutive natural numbers)

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

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 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 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.

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)

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"..

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 the exponent of the largest power of 2 whose digits are all distinct: 2²⁹ = 536 870 912

= 2² + 3² + 4² (sum of 3 consecutive squares)
= 3 + 5 + 7 + 11 + 13 (sum of consecutive primes)

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

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

29² = 21² + 20² (Pythagorean triple)

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.

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

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".

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

= 2⁵ - 1
= 2² + 3³
= 1 + 5 + 5²
= 2⁰ + 2¹ + 2² + 2³ + 2⁴ = 1 + 2 + 4 + 8 + 16
= 3³ + 3 + 3/3

31² x 325 = 312325

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.

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.

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

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

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:
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

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 two 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.

2 dice can fall in 36 different manners.

= 3! x 6
= 1³ + 2³ + 3³

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.

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

3727 is a prime number
37273 is a prime number
372733 is a prime number
3727333 is a prime number
37273339 is a prime number
372733391 is a prime number
3727333913 is a prime number
37273339139 is a prime number

2³⁷ = 137438953472
37! = 13763753091226345046315979581580902400000000

37 x 3 = 111; 37 x 6 = 222; 37 x 9 = 333; 37 x 12 = 444; etc.
37 x (6 + 6 + 6) = 666
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

If "abc" is a multiple of 37, then so are "bca" and "cab":

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.

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

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

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)

-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.

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.

is a sum of two consecutive squares: 4² + 5²

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 produces prime numbers for all integers n, from n=0 to n=40

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...
- 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.

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

= 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.

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 th