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.
A
free 197page eBook containing a huge collection of number
facts is available for
you to download.

If
you got a distinctive fact about any number listed here you think Archimedes'
Lab community might enjoy, why not post it here? 
Conoscete
un numero con delle proprietà originali? Contattateci! 
Connaissezvous
un nombre avec des propriétés étonnantes? Contacteznous! 
Number
list: lista
dei numeri (it), liste des nombres (fr), lista de números (es,
por), Liste besonderer Zahlen (ger), getallen en getalverzamelingen (du), seznam čísel (cz), 數表 (ch), 数の一覧 (jap), список чисел (ru), שמות מספרים (he).
06  712  1323  2469  70200  201786  5H0P
24 
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... >
= 2^{10} 
10^{3}
= 7^{2}  5^{2} (difference of squares)
= 4!
= 3 + 5 + 7 + 9 (sum of consecutive odd numbers)
= 2^{5} 
2^{3}
= 3^{3} 
3^{1}
= (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, 37^{2} 
1 = 1368 is divisible by 24. However, this works just
as well with any odd
number not divisible by 3. For instance, 125^{2} 
1 = 15624 is divisible by 24. In fact, any number of the
form p^{2}  q^{2} 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.
The Slinky
toy (aka "Lazy Spring") is a toy consisting of
a 24meter (18foot) 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!

25 
is
a centered
octagonal number.
is the smallest square number
that can be written as a sum of 2 (consecutive) squares:
5^{2} = 4^{2} + 3^{2
}is
an automorphic cubic
number: 25^{3} =
625
= 1
+ 3 + 5 + 7 + 9 (sum of the first five odd numbers)
= 3 + 4 + 5 + 6 + 7 (sum of consecutive numbers)
= 1^{2} +
2^{2 }+ 2^{2} + 4^{2}
= (1^{2} + 2^{2})^{2
}= 13^{2}  12^{2}
25^{2 }=
7^{2} + 24^{2} (Pythagorean
triple)
4! +
1 = 25
The
powers of 25 (or any other number ending by 25) end with 25:
725^{2} = 525625
When
you add the digit 6 to 25 you obtain two squares:
625 =
25^{2}, and 256 =
16^{2}
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)
17 
24 
1 
8 
15 
23 
5 
7 
14 
16 
4 
6 
13 
20 
22 
10 
12 
19 
21 
3 
11 
18 
25 
2 
9

(The
Magic Square of Mars is associated with the Sator
Square)
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: 5^{2 }+
2 = 3^{3}
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".

26 
is
the only number sandwiched between a square (5^{2})
and a cube (3^{3}).
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: 26^{2} = 676
is the sum of the following series (J. Bernoulli):
The
26 letters of the alphabet can make 40,329x10^{22} 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 26^{3} = 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 lettervalues 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
26letter 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.

27 
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 2digit 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:
1^{2} + 1^{2} + 5^{2} = 27 and 3^{2} +
3^{2} + 3^{2} = 27
is the
largest integer which is the sum of the digits of its cube:
27^{3} = 19,683 and 27 = 1 + 9 + 6 + 8 + 3
27^{3} =
3^{3} + 18^{3} + 24^{3}
27 =
3^{3}
27 = 6^{2}  3^{2} (difference
of squares)
27 = 14^{2} 
13^{2} (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 = 4p^{3} + 27q^{2}
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 IChing 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.
Twentyseven
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 10^{27}.
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
TwentySeven 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 twentyseven
boys and twentyseven girls.
Every March 16th and 17th, a solemn procession made a circuit of
twentyseven chapels called "Argei" located
in various parts of the city of Rome.
The
27letter 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.

28 
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 = 2^{2}(2^{3} 
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! / 360^{2}
= 8^{2 }– 6^{2} (difference
of squares)
= 2^{5 }– 2^{2} (difference
of powers of 2)
28
x 157 = 4396 (this equality contains all digits
1 to 9 once)
A
28sided 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
28letter 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!

29 
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 2n^{2} 
1 is a square, in fact: 2 x 29^{2}  1 = 41^{2} 
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:
x^{2} + y^{2} +
z^{2} = 3xyz
is the exponent of the largest power of 2 whose digits are all
distinct: 2^{29} = 536 870 91
is
the smallest multidigit prime whose product of digits of its cube
is also a cube:
29^{3} = 24389 and 2 x 4 x 3 x 8 x 9 = 1728 = 12^{3}
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
= 2^{2} +
5^{2 }(sum of 2 squares)
= 2^{2} +
3^{2} + 4^{2 }(sum
of 3 consecutive squares)
= (6!
+ (6! + 6)/6)
=
(2 x 9) + (2 + 9)
=
2x5 + 3x3 + 5x2 (palindromic expression)
29^{2 }=
21^{2} + 20^{2} (Pythagorean
triple)
29
+ n^{2} is prime for n = 0 to 291 (posted
by Len Baumann)
3^{29} 
2^{29} 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, nonleap 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:
1^{2} + 2^{2} + 3^{2} + 4^{2}
is the largest number such that every smaller coprime to
it is prime.
1^{1} +
2^{2} + 3^{3} ... + 28^{28} + 29^{29} +
30^{30} is prime!
 Crespi de Valldaura
= 4
+ 5 + 6 + 7 + 8
≈ π^{3}  π/500
30 + 25 =
55 and 55 x 55 = 3025
n^{5 }– 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) 
Indoeuropean
 *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] me^{o}be;
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. 

31 
is
the first lucky
prime, and the 3rd Mersenne
prime (2^{5}  1).
is very close to ^{3}.
= 2^{5} 
1
= 2^{2} + 3^{3}
= 1 + 5 + 5^{2}
= 2^{0} + 2^{1} + 2^{2} + 2^{3} +
2^{4} = 1 + 2 + 4 + 8 + 16
= 3^{3} + 3 + 3/3
31^{2} x
325 = 312325
the
sum of the first 31 odd primes is a perfect square:
3 + 5 + 7 + 11 + ... + 83 + 87 + 89 = 31^{2}
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 trenteetun
(31)" means 'to be dressed to the nines' in English, and "anar
de vintiun (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...)

32 
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: 2^{5}
= 2^{4 }+ 4^{2} (Leyland
number)
= 5 + 7 + 9 + 11
32!  1 is prime
(32 + 1)!  1 is also prime
log_{2} 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.

33 
is
a 12gonal
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.
= 0^{0} +
1^{1 }+ 2^{2 }+ 3^{3}
= 1^{5 }+
3^{3} + 5^{1}
= 17^{2}  16^{2} and 17 + 16 = 33
= 7^{2}  4^{2}
= 1^{5} + 5^{5}
= 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 = 16^{3 }+
50^{3} + 33^{3} and
1000165033 =
1000^{3} + 16^{3} +
50^{3 }+ 33^{3}
33^{2} =
65^{2}  56^{2} and
33 33^{2} = 65 65^{2}  56 56^{2}
33!
 1 is a prime number.
1123
is a divisor of 33^{3}  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: trentetrois,
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 smiletothecamera word, as is 'cheese' in English. "Diga
treinta y tres" ('say thirtythree') is the same
as 'say cheese'.

34 
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
4x4
Magic squares with a 'magic constant' of 34
One
of the 86 magic squares having a 'magic
constant' of 34:

1 
14 
4 
15 
12 
7 
9 
6 
13 
2 
16 
3 
8 
11 
5 
10 

Can
you spot why this panmagic
square is particularly special? 
15 
10 
3 
6 
4 
5 
16 
9 
14 
11 
2 
7 
1 
8 
13 
12 


35 
is
a tetrahedral number  the sum of the first five triangular numbers:
1 + 3 + 6 +10 + 15
= (10 – 3)(10 – 5) (multiplication
of its 10complement digits)
There
are almost 35 hexominoes,
the polyominos made from 6 squares.

36 
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.
= 3!
x 6
= 1^{3} + 2^{3} + 3^{3} = (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 mmddyyyy 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 GraecoLatin square. The answer is that no such
arrangement is possible...
36
Hidden Holy Persons
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 "Lamed Vavniks", or "Hidden
Tzaddikim", goes back to the early days of Chassidism, in
the 17th century.

37 
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
(HilbertWaring'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 6digit
number in which the first 3 digits are consecutivelyincreasing
numbers; and the last 3 digits, consecutivelydecreasing numbers,
is divisible by 37. For instance, the numbers 123987, 234765
and 567543 are all divisible by 37.
= 2^{2} +
2^{2} + 2^{2} + 5^{2}
= 4^{3}  3^{3}
37^{2} = 12^{2} + 35^{2 }= 19^{2} 
18^{2} (Pythagorean
triple)
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 10^{37} = 3 333 333 333 333^{3}
2^{37} =
137438953472
37! = 13763753091226345046315979581580902400000000
3^{2} +
7^{2}  3 x 7 = 37
3 x (3 + 7) + 7 = 37
(3^{3} + 7^{3})/(3^{ }+ 7)^{ }=
(4^{3}  3^{3})/(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
If a
3digit number of the form "ABC" is a multiple
of 37, then so are numbers "BCA" and "CAB":
37
x 1 = 037
37 x 10 = 370
37 x 19 = 703
37
x 2 = 074
37 x 11 = 407
37 x 20 = 740

37
x 4 = 148
37 x 13 = 481
37 x 22 = 814
37
x 5 = 185
37 x 14 = 518
37 x 23 = 851

37
x 7 = 259
37 x 16 = 592
37 x 25 = 925
37
x 8 = 296
37 x 17 = 629
37 x 26 = 962

(multiplication
with multiples of 3 see previous note) 
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!
Dudeney's
prime magic square
67 
1 
43 
13 
37 
61 
31 
73 
7 
'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.

38 
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 = 2^{2} + 3^{2} + 5^{2}
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)
There
is only one possible magic
hexagon (with magic constant 38)




3 






17 



19 


18 



7 



16 


1 



2 


11 



5 



12 


6 



4 


9 



8 



10 


14 



13 






15 




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.

39 
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
= 20^{2} 
19^{2} = 20 + 19
= 8^{2}  5^{2} (difference
of squares)
= 3^{1} + 3^{2} +
3^{3} (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: 392333.
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 directdialed phone calls to Italy.

40

is
a 8gonal number, and as the sum of the first four pentagonal
numbers (1, 5, 12, 22), it is also a pentagonal pyramidal number.
= 2^{3} x
5
= 6 + 7 + 8 + 9 + 10 (sum of consecutive numbers)
= 3^{0} + 3^{1} + 3^{2 }+ 3^{3} (sum
of powers of 3)
= 2^{3} + 2^{3 }+ 2^{3} + 2^{3} +
2^{3} (sum of cubes)
= (41^{2} 
9^{2})
= (3^{4}  1)/2 (number of the form (3^{n} 
1)/2)
40^{40} 
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 socalled ‘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 fortyday 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, selfdenial, 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.
WD40 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. WD40 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.
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) 
Indoeuropean
 *K^{w}ETURKOMT(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. 

41 
is
a sum of two consecutive squares: 4^{2} + 5^{2}
is the
lowest number whose cube is the sum of 3 cube numbers in 2 different
ways:
41^{3} = 2^{3} + 17^{3} + 40^{3} =
6^{3 }+ 32^{3} + 33^{3} = 68,921
is the
smallest nonpalindromic prime which on subtracting its reverse
gives a perfect cube (41  14 = 3^{3}), and is also the
smallest number that is not of the form 2^{x}  3^{y}.
n^{2}  n + 41 produces
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...

posted by Ryan Bartling 

42 
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:
42^{2} = (11^{3} + 13^{3})/2
= 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)
= 2^{2} + 2^{2} + 3^{2} + 5^{2} (sum
of four prime squares)
= 6^{1} + 6^{2} = 7^{2}  7^{1} (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 n^{7}  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.

43 
is
the smallest nonpalindromic prime which on subtracting its reverse
gives a perfect square: 43  34 = 3^{2}
is the
number of sided 7iamonds (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 ...
= 4^{2} + 3^{3}
= 22^{2}  21^{2} (difference of
two consecutive squares)
= 2^{3} + 2^{3} + 3^{3
}= 6^{0} + 6^{1} + 6^{2} (sum
of powers of 6)
43^{7} =
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 +
4^{3} 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 'fortythree'
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 citrusvanilla
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
IChing hexagram: ¦ is named 夬, guài ('parting')
and means resolution, breakthrough.
Londinium (now
London) was founded by the Romans in AD 43.
It
requires 43 muscles to frown.
"This
morning, I wrote a cheque for € 43million.
That's what I paid for the last fiscal year"  Silvio
Berlusconi, former Italian prime minister. June 15, 2007.

44 
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...
=
12^{2}  10^{2}
= !5 = 5! (11/1!+1/2!1/3!+1/4!1/5!)
In
the following equations x=44:
x^{2} + y^{2} = a^{2}
y^{2} + z^{2} = b^{2}
z^{2} + x^{2} = c^{2}

45 
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) + (102) + (10x2)
+ (10/2)
= 3^{2} + 6^{2}
= 6! / 2^{4} = 6! / 4^{2}
45^{2} =
2025
20 + 25 = 45
45^{3} = 91125
9 + 11 + 25 = 45
45^{4} = 4100625
4 + 10 + 06 + 25 =
45
45^{3} =
54^{2} + 297^{2} = 135^{2} + 270^{2} (sums
of 2 squares)
is a
graphical selfreference number in French:
QU4R4NTE+C1NQ

46 
is
the number of different possible arrangements of 9 nonattacking
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.

47 
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.
= 2^{5} +
5^{2}  2 x 5
is selfconscious (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
= 24^{2} 
23^{2} = 24 + 23
4^{3} + 7^{3} = 407
666^{47} has a digit sum of 666.
9999999999999999999999999999999999999999999999 =
47
212765957446808510638297872340425531914893617
is
a cult number; some people claim that the number fortyseven
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 fiftyseven sheets of toilet paper
each day.
Buy
your favorite
Number (47) here.

48 
is
the smallest number with 10 proper divisors. Actually, 48
= 2^{4} 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 17gonal number, and
a Harshad
number.
is an arccotangent
irreducible number, that is a positive integer n for
which the greatest prime factor of n^{2} + 1 meets
or exceeds 2n: 48^{2} + 1 = 5 x 461 and 461 > 2
x 48
The product of all 10 divisors of 48 equals 48^{4}.
=
13^{2}  11^{2} = 8^{2}  4^{2} =
7^{2}  1^{2}
= (4^{3} + 8^{3}) / (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)
48^{2} =
2^{2} + 5^{2 }+ 8^{2 }+ 11^{2 }+
14^{2 }+17^{2 }+ 20^{2 }+ 23^{2} +
26^{2} (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 fortyeight) means all hell breaks  or will
break  loose. While "mandare qualcuno a carte quarantotto" (to
send someone to cards fortyeight), 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"

49 
is
the smallest number with the property that it and its bordering
numbers are squareful:
48 = 4^{2} x 3, 49 = 7^{2 }x 1, 50 = 5^{2} x
2
is a
perfect square: it is the smallest number which is the concatenation
of two prime squares (2^{2} and 3^{2}) 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: 49^{3} = 117,649
2 fourth
powers sharing exactly the same digits:
49^{4} = 5,764,801 and 32^{4} = 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)
= 7^{2} = 25^{2}  24^{2} = 25 + 24
= (11  4)^{(11
 9)}
= 3^{2} + 5^{2} + 3 x 5
= 3^{2} + 10 x 2^{2}
= 5^{2} + 6 x 2^{2}
49
= 4 + 9
Illicit
semplification:
49 = 49 = 4
98 98 8
An algebraic
expression that is always divisible by 49:
2^{3n+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 'fortynine'
as XLIX, not as the 'shortcut' IL.
During
the GoldRush, the goldseekers that came to California were
called 'fortyniners' 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, hydroxyphenyl2butanone (10 percent
solution in alcohol), aionone, isobutyl anthranilate, isobutyl
butyrate, lemon essential oil, maltol, 4methylacetophenone, 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, gammaundecalactone, vanillin,
and solvent.
Buy
your favorite
Number (49) here.

50
L 
is
the smallest number that can be written as the sum of 2 squares in
2 distinct ways: 1^{2} + 7^{2} and 5^{2} +
5^{2}
is also the sum of 3 squares: 3^{2} + 4^{2} + 5^{2}
and the product of complex
numbers: (7 + i) (7  i) =
= 49 + 7i  7i  i^{2} = 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
5050/90 rule: Anytime you have a 5050 chance of getting
something right, there's a 90% probability you'll get it wrong.
FIFTY in
different languages (© G.
Sarcone)
(Old English fîftig) 
Indoeuropean
 *PENK^{w}EKOMT(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  PE^{n}TIDESETJ  
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 negyven.
Mayan lahunyoxkal (10 to 60).
Nahuatl ōmpōhualmahtlāctli (2
x 20 and 10).
Suomi viisikymmentä.
Zulu amashumi amahlanu. 

51 
is
a pentagonal number as well as a centered pentagonal number (one
of the few numbers to be both) and an 18gonal 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
units^{2}). 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)
= 26^{2}  25^{2} = 26 + 25
= 2^{3} + 2^{3} + 2^{3} + 3^{3} (sum
of cubes)
is the
'AntiDevil' number.
According to Abellio, 51 symbolizes the "Universal Time".
"51%
is not a mandate!" is an antiBush 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.
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.
Buy
your favorite
Number (51) here.

52 
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 4^{2}  3 x 4).
is a centered 17gonal 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).
= 5^{2} +
3^{3}
= 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 nonconnected 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 welluttered speech of one who acts accordingly.
 Buddha (240 B.C.)

53 
is
the smallest prime which is not the difference between powers
of 2 and 3.
is the only twodigit number that is reversed in hexadecimal (base
16): 35_{16}
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
= 2^{2} + 7^{2} = 1^{2} + 4^{2} +
6^{2} (sum of squares)
53^{2} = 28^{2} + 45^{2}
53^{5} =
418,195,493 = (4+18+1+9+5+4+9+3)^{5}
53^{7} = 1,174,711,139,837 = (1+1+7+4+7+1+1+1+3+9+8+3+7)^{7}
2^{53} =
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.

54 
is
a 19gonal number.
is the smallest number that can be written as the sum of 3 squares in
3 different ways:
7^{2} + 2^{2} + 1^{2} = 6^{2} +
3^{2} + 3^{2} = 5^{2} + 5^{2} +
2^{2}
= 3^{3} +
3^{3} = = 3^{4}  3^{3}
= (3^{3} + 3^{2} + 3^{1} + 3^{0})
+ (3^{3 } 3^{2}  3^{1}  3^{0})

55 
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:
= 1^{2} + 2^{2} + 3^{2} + 4^{2} +
5^{2}
=
3 + 6 + 10 + 15 + 21 (sum of consecutive triangular
numbers)
=
28^{2}  27^{2} = 28 + 27
= 8^{2}  3^{2}
= 2^{0} +
2^{1} + 2^{2} + 2^{3} + 3^{0} +
3^{1} + 3^{2} + 3^{3
}= 10^{2} 9^{2} + 8^{2} 7^{2} +
6^{2} 5^{2} + 4^{2} 3^{2} + 2^{2} 1^{2}
5^{3} +
5^{3} = 250 and 2^{3} + 5^{3} + 0^{3} = 133
and 1^{3} + 3^{3} + 3^{3} =
55
55^{2} =
44^{2} + 33^{2
}55^{2} = 3025
and 30 + 25 = 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.
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 "73855 rule" states
that ommunication is 7% verbal, 38% tone of voice, and 55%
body language.

56 
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)
56^{2} =
65^{2}  33^{2}
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}=10^{100}.
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.

57 
=
3 x 19
= 111 in base 7 (that is: 57 = 7^{2} + 7^{1} +
7^{0})
= 7 + 8 + 9 + 10 + 11 + 12
= 1^{2} + 2^{2} + 4^{2} + 6^{2} (sum
of squares)
= 29^{2}  28^{2} = 29 + 28
is a 20gonal number.
can be expressed as a sum x^{y} + y^{x},
where x and y are integers greater than 1: 57
= 2^{5} + 5^{2}.
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 2375757, and the address
is P.O. Box 57. "Heinz 57" is also used as a joke slang
term for mixedbreed 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.

58 
is
the sum of the first seven prime numbers:
2 + 3 + 5 + 7 + 11 + 13 + 17
=
13 + 14 + 15 + 16 (sum of consecutive numbers)
= 2^{2} + 2^{2} + 5^{2} + 5^{2
}= 3^{2} + 7^{2}^{ }
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.

59 
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)
= 30^{2}  29^{2} = 30 + 29
= 2^{3} + 2^{3} + 2^{3} + 2^{3} +
3^{3 }(sum of cubes)
59^{4} =
133^{4} + 134^{4}  158^{4 }(Euler argued
that there were solutions to the equation a^{4}+ b^{4} = c^{4}+ d^{4},
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.

60 
is
an abundant
number.
is adjacent to 2 prime numbers: 59 and 61.
is also an heptagonal
pyramidal number and a 21gonal
number (icosihenagonal number of the form (19n^{2} 
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)
= 8^{2}  2^{2} = 16^{2}  14^{2} (difference
of squares)
= 2^{6}  2^{2} (difference of powers
with same base)
= 4 x 4 x 4 – 4 = 44 + 4 x 4 = 4^{4}/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 ddmmyyyy 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.
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) 
Indoeuropean
 *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  S^{h}ESTIDESETJ  
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. 

61 
is
a cuban prime of
the form p = (x^{3}  y^{3})/(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.
= 5^{3 }
4^{3}
= 5^{2 }+ 6^{2
}= 31^{2}  30^{2} = 31 + 30
61^{2} =
11^{2} + 60^{2} (Pythagorean
triple)
is the smallest prime such that its square can be expressed as
the sum of 1, 2, 3, and 4 distinct squares:
61^{2} = 60^{2} + 11^{2} = 52^{2} +
24^{2} + 21^{2} = 56^{2} + 22^{2} +
10^{2} + 1^{2}
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 inches^{3}
“61*” is
also the name of TV show/series or movie.

62 
is
the smallest number that can be written as the sum of of 3 distinct squares in
2 distinct ways:
= 1^{2} + 5^{2} + 6^{2}
= 2^{2} + 3^{2} + 7^{2}
= 2^{3} +
3^{3} + 3^{3} (sum of cubes)

63 
=
2^{6 }– 1
= 6^{2} + 3^{3}
is the number of partially
ordered sets of 5 elements.
is a selfreference number:

64 
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:
2^{6} = 4^{3} = 8^{2 } = 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)
= 17^{2}  15^{2}
= 10^{2}  6^{2}
= 2^{7}  2^{6}
= (2^{2} + 2^{2})(2^{2 }+ 2^{2})
= 4
= (2 +/ 2i3)^{3} =
(1 +/ i3)^{6} (power
of complex numbers)
= 7
x 3^{2} + 1^{2 }(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 2^{64} + 1 (≈ 1.84467441
x 10^{19}) 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.

65 
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.
= 1^{5} +
2^{4} + 3^{3} + 4^{2} + 5^{1 } 
posted by Jim O'Donohoe
= (63^{2} +
16^{2})
= (56^{2} +
33^{2})
= 1^{2} + 8^{2
}= 4^{2} + 7^{2}
= 33^{2}  32^{2}
= 2 + 3 + 4 + ... + 9 + 10 + 11
65 + 56
= 121 = 11^{2}
65 – 56
= 9 = 3^{2
}65^{2} – 56^{2} =
33^{2 } (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.

66 
is
a triangular palindromic number and the number of 8iamonds.
= 1
+ 2 + 3 + … + 10 + 11
= 2^{2} + 2^{2} + 3^{2} + 7^{2} (sum
of square numbers)
Is the
largest number <100 that contains no letter 'e' (sixtysix).
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.

67 
is
the smallest number which is both palindromic in
base 5 or 6.
is the sum of five consecutive primes: 7 + 11 + 13 + 17 + 19
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.

68 
is
the smallest composite number that becomes prime by turning it
upside down.
= 1
x 2^{2} 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 nearrevolutionary 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.

69 
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 n^{2} and n^{3} all
together contain each digit once: 69^{2} = 4,761 and
69^{3} = 328,509
is often
the largest factorial that can be calculated by a pocket calculator:
69!≈1.71x10^{98} (less than 10^{100})
69^{32} =
6.96969696 · 10^{58}
69^{2} = 4761
69^{3} =
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.
Buy
your favorite
Number (69) here .

06  712  1323  2469  70200  201684  5H0P
You're
encouraged to expand and/or improve the number facts featured
on this page. Send
us your comments, feedback or suggestions.
You can use any of our content (pictures, texts) in your noncommercial
web page for free as long as you leave the copyright and author's info intact
and link to us.
This
work is licensed under a Creative
Commons Attribution  NonCommercial  ShareAlike 2.5 License.
Source
credits
1. We always credit our sources, when we can.
2. Some of the material on this web section is submitted by our
readers/visitors. Where cited, excerpts are quoted or italicized,
and authorship credit is prominently supplied.
3. Most authors/submitters, however, prefer to remain anonymous,
for reasons of their own, and we respect their wishes.
4. If you believe you have identified material on this web site
that has not proper permission (and can provide us with verifiable
proof of ownership), or that should (but does not) credit a verifiable
source, please bring it to our
attention.
5. Any oversights, omissions and corrections will be updated as
soon as possible.
Encyclopedia
of Numbers © G. Sarcone, Archimedes Lab,
Genoa, Italy

