70 
is
a Pell
number and the smallest abundant
number which is not the sum of some subset of its divisors.
=
7 + 8 + 9 + 10 + 11 + 12 + 13
= 2(6^{2} 1^{2}) = 2(6 + 1)(6 1)
70^{2} =
1^{2} + 2^{2} + 3^{2} + 4^{2} +
... + 22^{2} + 23^{2} + 24^{2}
Symbolism:
70 corresponds to the totality of an evolution, an evolutionary
cycle being fully completed, according to Saint Augustin.
The corresponding Hebraic letter is 'ain, which corresponded
to the sixteenth mystery of the Tarot: the houseGod, symbol of
fall, collapse, catastrophe  there is agreement with the idea
of termination of cycle, according to R. Allendy.
In the Bible:
The 70 weeks prophecy of the book of Daniel. (Dn 9,24)
The 70 years of captivity of Babylonian of two of the twelve tributes
of Israel. (2 Ch 36,21)
Ezekiel sees 70 ancients adoring the idols. (Ezk 8,11)
70
liters of water are needed to produce just 1 apple!
SEVENTY in
different languages (© G.
Sarcone)
(Old English hundseofontig)

Indoeuropean
 *SEPT'MKOMT(A) 
Sanskrit  saptati 
Greek, Attic  ἑβδομήκοντα HEBDOMEKONTA 
Latin  SEPTUAGINTA  
Italian settanta;
French soixantedix (= 60 + 10;
Swiss French and Belgian septante);
Spanish and Portuguese setenta;
Catalan and Provençal setanta;
Romanian şaptezeci; Rumansh siatonta;
Sardinian setànta. 
Old
Germanic  *SIBUNTEHUND  
Dutch zeventig;
German siebzig; Danish halvfjerds (halvfjerdsindstyve archaic
= 3 twenties + 1/2 of a fourth twenty); Norvegian sytti;
Swedish sjuttio; Icelandic sjötíu. 
Old
Slavic  SEDMIDESETJ  
Russian семьдесят sem'desyat;
Czech sedmdesát; Slovenian sedemdeset;
Polish siedemdziesiąt. 
Chinese 七十 qīshí (complex
form 柒拾). 
Ancient
Egyptian [sfḫy.w] sefkheyu;
Akkadian sebeā. 
Arabic سبعون sab’un;
Hebrew שבעים sheeveem;
Maltese: sebgħin; Amharic sèba. 
More
languages
Magyar hetven.
Mayan lahucankal (80 – 10).
Nahuatl yēmpōhualmahtlāctli.
Suomi seitsemänkymmentä.
Zulu amashumi ayisikhombisa. 

71 
is
the first of a pair of twin
primes: 71, 73. Twin primes, such as 3 and
5, are primes that differ by 2 (i.e. the smallest possible difference
between two primes).
divides
the sum of the antecedent primes,
in fact:
2 + 3 + 5 + 7 + 11 + ... + 67 + 71 is divisible by 71.
= (4!
+ 4.4)/.4 (representation of numbers using only four
4's)
= 36^{2}  35^{2} = 36 + 35
71 
1 = 1 x 2 x 5 x 7 and 71 + 1 = 3 x 4 x 6 (products
of partitions of consecutive numbers)
71^{2} =
7! + 1
71^{2} = 2^{7} + 17^{3 }(sum
of prime powers of two prime numbers)
71^{3} = 357911 (consecutive
odd numbers)
71^{4} =
136^{3} +
4785^{2}

72 
is
the smallest number that can be written as a sum of two primes,
each ending with the digit 1, in two distinct ways:
72 = 11 + 61 = 31 + 41
is the largest number of spheres that can touch another sphere in
a lattice packing in 6 dimensions.
Dividing 72 by the interest rate (x percent),
you obtain the number of years you should wait to double
your initial capital.
= 2
+ 4 + 6 + 8 + 10 + 12 + 14 + 16 (sum of consecutive
even integers)
= 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 (sum of consecutive
integers)
=
5 + 7 + 11 + 13 + 17 + 19 (sum of 6 consecutive
primes)
= 13 + 17 + 19 + 23 (sum of 4 consecutive primes)
= (1
x 2 x 3 x 4 x 5 x 6)/(1 + 2 + 3 + 4)  G.
Sarcone
= 2^{3}(2^{3} +
1) = 3^{2}(3^{2} 
1)
= 2^{3} · 3^{2
}=
11^{2}  7^{2} = 19^{2}  17^{2 }(difference
of 2 consecutive squared primes)
= 2^{3} + 4^{3} = 3^{4}  3^{2}
72
= 24 x 3 and reversely 27 = 24 + 3
72^{5} =
19^{5 }+ 43^{5} + 46^{5} + 47^{5} +
67^{5}
The
isosceles triangle having a pair of 72° angles
is called a Golden
Triangle.
The Great
Rhombicuboctahedron (see opposite) is an Archimedean
Solid with 72 Edges.
The
pulse rate of normal adult human is about 72 beats per
minute at rest.
72
hours = 3 days
The
72 names of God in the Kabbalistic tradition.
Incredible
geometric sculpture made with 72 pencils by
the artist and mathematician George
W. Hart. [click
here to enlarge]

73 
is
a sexy
prime and an emirp.
is the smallest prime whose digits are reversed
in base 22 and the smallest number (besides 1) which is one less
than the double of its reverse: 2 x 37  73 = 1
is the first difficult prime to form under the rules of the four
4's puzzle (The goal of the puzzle is to find the simplest
mathematical expression for every integer, using only common mathematical
symbols and the digit four).
is a divisor of 10^{4} + 1 = 10001 (composite
Fermat number).
Every
positive integer can be expressed as a sum of 73 or fewer sixth
powers.
The
mirror of 73  the 21st prime number  is 37  the 12th prime
number.
The
sum of the first 73 odd primes is divisible by 73.
73 =
37^{2 }– 36^{2} =
37 + 36
73^{2 }=
48^{2} + 55^{2} (Pythagorean
triple).
365
days in a year : 5 = 73
73 
= 
14 
+ 
21 
+ 
13 
+ 
2 
+ 
5 
+ 
18 


N 

U 

M 

B 

E 

R 
(each
number of the sum represents a letter in its alphabetical
order) 
73 =
111_{8}
73
= (9! + 7! ) / 7!
is
the smallest number with the property that it and its neighbors
can be written as a sum of 2 squares.
72 = 6^{2} + 6^{2
}73 = 3^{2} + 8^{2}
74 = 5^{2} + 7^{2}
 Found by Daniel Clemente from
the Polytechnical University of Catalonia
Most
buses in fiction novels display the route number 73. Actually,
many writers and authors chose 73 as the "archetypal" bus
route.
Buy
your favorite
Number (73) here.

74 
is
a nontotient.
is the number of nonHamiltonian polyhedra
with a minimum number of vertices.
= 17
+ 18 + 19 + 20
74%
is the storage density of a volume filled with small spheres.

75 
is
the number of orderings of 4 objects with ties allowed.
is a repfigit
number...
If you
take the "digit sum" 7+5=12 and then continue adding
in twos in "fibonacci style" look what happens:
7 + 5 = 12
5 + 12 = 17
12 + 17 = 29
17 + 29 = 46
29 + 46 = 75 ...arriving back at 75!
75^{5} =
19^{5} + 43^{5} + 46^{5} + 47^{5} +
67^{5}
75
liters of water are needed to produce 1 glass of
beer (250 ml)!
A dollar
bill contains 75% cotton and 25% linen.

76 
is
an automorphic
number:
76^{2} = 5776
76^{3} = 438976
76^{4} = 33362176
76^{5} = 2535525376...
= 8
+ 13 + 21 + 34 (sum of consecutive Fibonacci
numbers)
= 3^{2} +
3^{2} +
3^{2} +
7^{2} (sum
of square primes)

77 
is
the largest number that cannot be written as a sum of distinct
numbers whose reciprocals add up to 1. For example:
2 + 6 + 8 + 10 + 12 + 40 = 78 and
1/2 + 1/6 + 1/8 + 1/10 + 1/12 + 1/40 = 1
is the number of integer partitions of 12.
is the sum of 3 squares:
4^{2} + 5^{2} + 6^{2}
is the sum of the first 8 prime numbers:
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19
=
1001 / (77  37  27)  By G. Sarcone
= 39^{2}  38^{2}
= 9^{2}  2^{2}
77^{2} =
18^{2} + 19^{2 }+ 20^{2} + … +
26^{2} + 27^{2} + 28^{2 }(sum
of consecutive square)
8 x
8 + 13 = 77
88 x 8 + 13 = 717
888 x 8 + 13 = 7117
8888 x 8 + 13 = 71117
etc...
Any
number having a abcabc numeral
pattern is divisible by 77 (replace each letter with a digit.
E.g. 123123)
Seventyseven
is the smallest number requiring five syllables in English.
French
speaking people have two different names for seventyseven: 'Soixantedixsept'
(sixty and seventeen) and 'septantesept' (Swiss French
people).
During
World War II in Sweden at the border with Norway, "77" (sjuttiosju)
was used as a 'Shibboleth' (discriminatory password), because
the tricky pronunciation in Swedish (approx. 'shütioohshü')
made it easy to instantly discern whether the speaker was native
Swedish, Norwegian, or German.
Renovation
of a landmark Los Angeles restaurant has revealed a neon lamp
that has been left on for about 77 years! Source: Dailymail.
Conspiracy
theorists claim that the number 77 appears
statistically in many mass death, e.g., Flight 77 crashing
into the Pentagon on September 11, or the infamous 7 July 2005
London bombings, also known as 7/7.

78 
is
the lowest number which can be written as the sum of of 4 distinct squares in
3 ways.
is a triangular number (sum of consecutive integers
from 1 to 12):
78 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
= 37
+ 41 (sum of consecutive primes)
78^{3} =
39^{3} + 52^{3} + 65^{3}
Front
loaded interest is often offered on automobile loans. Front loaded
interest is also referred to as the Rules of 78 because
your total interest costs for the loan are divided into "78
parts" (sum of the digits of 12 months of a year). Those
78 parts are then paid during the first 12 months of the loan
according the following equation:
12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 78
The first month, you pay 12 of the 78 parts of the interest;
The second month, you pay 11 of the 78 parts;
The third month, you pay 10 of the 78 parts;
... and so on until the twelfth payment of 1 part interest.
The 78 RPM
phonograph records were the standard in 1926.
The common name for such records as 78s (or "seventyeights")
as they played at 78 revolutions per minute.

79 
is
a permutable prime,
and the smallest prime whose sum of digits is a 4th power.
is a Pillai
prime. A Pillai prime is a prime p for which there
is an integer n > 0 such that the factorial of n is
1 less than a multiple of the prime, but the prime is not 1 more
than a multiple of n, in short: n! ≡ 1
(mod p) but p ≠ 1 (mod n)
79 =
11 + 31 + 37 and
97 = 11 + 13 + 73 (sums of reversible primes)
=
(7 x 9) + (7 + 9) (this also works for 19, 29,
39, 49, ...)
=
40^{2}  39^{2} = 40 + 39
= 2^{7}  7^{2}
= 2^{2} + 5^{2} + 5^{2} + 5^{2} (sum
of squares)
≈ 8.888^{2}
x^{2}  79x
+ 1601, if you replace x in the previous expression with any
number from 1 to 79, you always obtain a prime number!

80 
is
the product of 5 primes.
is the smallest number with exactly 7 representations as a sum
of three distinct primes:
80 = 2 + 5 + 73 = 2 + 7 + 71 = 2 + 11 + 67 = 2 + 17 + 61 = 2 +
19 + 59 = 2 + 31 + 47 = 2 + 37 + 41
is the smallest integer n where n and n+1
are both products of 4 or more primes:
80 = 2 x 2 x 2 x 2 x 5 and 81 = 3 x 3 x 3 x 3
=
4 scores
= n x
2^{(n1)} = 5 x 2^{4}
= 3
+ 5 + 7 + 9 + 11 + 13 + 15 (sum of consecutive
odd numbers)
= 7 + 73 = 13 + 67 = 19 + 61 = 37 + 43 (4
distinct sums of 2 primes)
= 2^{3} + 2^{3} + 4^{3 }(sum
of 3 positive cubes)
= 4^{2} +
8^{2} = 4^{2} + 4^{3
}= 9^{2}  1^{2} = 292^{2}  44^{3}
= 6!
/ 3^{2}
80^{2} =
4^{3} + 8^{3} + 12^{3} + 16^{3}
is the
smallest number that is diminished by summing the alphabetical
order of its letters (writing out its English name and adding
the letters using a = 1, b = 2, c = 3, etc.): EIGHTY = 5 + 9
+ 7 + 8 + 20 + 25 = 74.
The "80:20
rule" originated from Vilfredo Pareto, an Italian economist.
He discovered a common phenomenon: about 80% of the wealth in
most countries was controlled by a consistent minority  about
20% of the people.
In French
there are 3 ways to say 80:
'quatrevingts', in France, and Quebec,
'huitante', in Switzerland,
and 'octante', in Belgium.
Spring
begins on March 21st, the 80th day of the year.
1 mile
= 8 furlongs = 80 chains
According
to U.S. statistics, an American woman would kiss an average of 80
men before she gets married.
"Around
the World in Eighty Days" (French: Le
tour du monde en quatrevingts jours) is a classic adventure
novel by the French writer Jules Verne.
EIGHTY in
different languages (© G.
Sarcone)
(Old English hundeahtatig)

Indoeuropean
 *OKTÔKOMT(A) 
Sanskrit  aśîti 
Greek, Attic  ὀγδοήκοντα OGDOKONTA 
Latin  OCTOGINTA  ; Medieval Latin  OCTUAGINTA  
Italian ottanta;
French quatrevingts (= 4 x 20;
Swiss French and Belgian huitante, octante 
archaic); Spanish ochenta; Portuguese oitenta;
Catalan vuitanta; Provençal ochanta;
Romanian optzeci; Rumansh otgonta;
Sardinian otànta. 
Old
Germanic  *AHTÔTEHUND  
Dutch tachtig;
German achtzig; Danish firs (firsindstyve archaic
= 4 times twenty); Norvegian åtti;
Swedish åttio; Icelandic áttatíu. 
Old
Slavic  OS(T)MIDESETJ  
Russian восемьдесят vosem'desyat;
Czech osmdesát; Slovenian osemdeset;
Polish osiemdziesiąt. 
Chinese 八十 bāshí (complex
form 捌拾). 
Ancient
Egyptian [ḫmny.w] khmeneyu;
Akkadian samānā. 
Arabic ثمانون thamanun;
Hebrew שמונים shmo'neem;
Maltese: tmenin; Amharic sèmanya. 
More
languages
Magyar nyolcvan.
Mayan cankal (4 x 20).
Nahuatl nāuhpōhualli.
Suomi kahdeksankymmentä.
Zulu amashumi ayisishiyagalombili. 

81 
is
a Heptagonal
number, that is a polygonal number of the form n(5n 
3)/2
is the square of
the sum of its digits; there are no other numbers with this property
except 0 and 1.
is the 3rd lowest prime power of a prime power of a prime.
= 5
+ 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 (sum of consecutive
numbers)
= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 (sum
of consecutive odd numbers)
=
3^{4} = 9^{2}
= 2^{5} + 7^{2}
= 41^{2}  40^{2} = 41 + 40
= 1^{2} + 4^{2} + 8^{2}
=
(1 + 2 + 2 + 4)^{2} = 1^{3} + 2^{3} +
2^{3} + 4^{3}
x^{2}  81x
+ 1681, if you replace x in the previous expression with any
number from 1 to 80, you always obtain a prime number!
The
reciprocal of 81 gives an interesting recurring
decimal expansion:
1/81 = 0.123456790123456790123456790...
The result is actually a repeating decimal containing every digit
from 0 to 9, except for 8.  Posted by Ting Wang
However, the smallest fraction that gives a repeating
set of all the digits 09 placed successively is:
13717421/111111111
= 0.1234567890123456789...
Group
consecutive numbers like this:
1  2, 3  4, 5, 6  7, 8, 9, 10  11, 12, 13, 14, 15  ...
Then cancel every second group. Now, if you sum the first remaining n groups,
you will obtain n^{4}. For instance:
(1) + (4 + 5 + 6) + (11 + 12 + 13 + 14 + 15) = 3^{4} =
81
 Source: "Scripta Mathematica", Juzuk,
1939.
The Tao
Te Ching (道德經), the Chinese
book of wisdom, is a philosophical text of around 5,000 Chinese
characters in 81 brief chapters.
Turkish
pronunciation of 81 (seksen
bir) is exactly 'sexNbeer', this is a kind
of amusing wordplay for people who understand both Turkish and
English.
81,
is the number of squares on a Shogi board
(Japanese chess board).

82 
is
an happy
number, and a companion Pell
number.
is a 15gonal number and a centered 27gonal number.
is the number of different ways you can arrange 6 regular hexagons
by joining their adjacent sides; thus it is the number of possible 6hexes (with
82 6hexes it is possible to form an hexagonal
ring).
= 1 + 5 + 21 + 55 (sum of Fibonacci
numbers)
= 19 + 20 + 21 + 22 (sum of consecutive integers)
= 1^{2} + 9^{2
}= 2^{2} + 2^{2} + 5^{2} + 7^{2} (sum
of square primes)^{
}= 2(4^{2} + 5^{2})
Can
be expressed as 5 different sums of two primes:
82 = 3 + 79 = 11 + 71 = 23 + 59 = 29 + 53 = 41 + 41
(cf. Goldbach's
conjecture)
is
one of the nuclear
magic numbers in physics. 'Nuclear magic numbers' are
the numbers of nucleons (protons or neutrons) forming full outer
shells in an atomic nucleus, that are relevant for nuclear isotope
stability.
is the international telephone dialing code for Korea.
is the International Standard Book Number for books published in
Norway (ISBN 82...).
The verb to pay is used 82 times in the Bible.
The Latin
and Greek prefixes "duooctoginta", "dioctaconta", "dooctaconta", "octacontadi",
and/or "octacontakaidi" all mean 'eightytwo'.
The noun and adjective "duooctogintennary" means 'a group
of 82' or 'a 82year period'. The adjective and noun "duooctogintennial" means
'recurring every 82 years' or 'lasting 82 years'. The noun and
adjective "duooctogintennium" means 'a period of 82 years'.
The number
82 appears frequently throughout the film "Magnolia",
written and directed by Paul Thomas Anderson, and starring
Tom Cruise.
82nd
Verse in Tagore's Gitanjali:
Time is endless in thy hands, my Lord. There
is none to count thy minutes.
Days and nights pass and ages bloom and fade like flowers. Thou
knowest how to wait.
Thy centuries follow each other perfecting a small wild flower.
We have no time to lose, and having no time we must scramble for
a chances. We are too poor to be late.
And thus it is that time goes by while I give it to every querulous
man who claims it, and thine altar is empty of all offerings to
the last.
At the end of the day I hasten in fear lest thy gate to be shut;
but I find that yet there is time.

83 
is
the number of zeroless squares that are composed of the digits 1
up to 9.  Beilier
= 11
+ 31 + 41 (sum of the first 3 primes ending with
1)
it seems
there are over 10^{83} elementary particles with nonzero
mass in the known universe.

84 
is
a tetrahedral
number:
84 = 1 + 3 + 6 + 10 + 15 + 21 + 28 (sum of the first
7 triangular numbers)
is the
highest order of a permutation on
14 elements.
is the
smallest integer with 8 different representations as a sum of
2 primes:
84 = 5 + 79 = 11 + 73 = 13 + 71 = 17 + 67 = 23 + 61 = 31 + 53 =
37 + 47 = 41 + 43
12 x 84 =
48 x 21 (palindromic equality)
84 =
14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 = 6 x 14
= 2^{2} +
4^{2} + 8^{2} = 4^{1} + 4^{2} +
4^{3}
= 2^{5} +
3^{3} + 5^{2} (is
the smallest number that can be expressed as the sum of 3 distinct
primes raised to distinct prime exponents)
= 3^{2} + 5^{2} + 5^{2} + 5^{2} (sum
of cubes)
N = n^{7}  n, N is
divisible by 42, 84, 168 when n is
odd.
1000
+ 84 = 1084 (one thousand eightyfour) is the
smallest natural number whose name contains the five vowels a,
e, i, o, u in order.
Uranus
revolves around the Sun once every 84 Earth
years.
There
is a town named EightyFour in
Pennsylvania, USA.

85 
is
the largest known n for which 1^{2}+2^{2}+3^{2}+...+n^{2} =
1+2+3+...+m has a solution.
is a centered triangular number
.
85/38
is a good approximation of 5.
= 43^{2} – 42^{2} =
43 + 42
= 2^{2} + 9^{2}
= 6^{2} + 7^{2} (sum of 2 consecutive squares)
= 4^{0} + 4^{1} + 4^{2} + 4^{3} (sum
of powers of 4)
85^{2} =
13^{2} + 84^{2} = 36^{2} + 77^{2}

86 
is
an happy
number.
is the largest known n for which 2^{n} contain
NO zeros:
2^{86} = 77,371,252,455,336,267,181,195,264
(26 digits).
= 222
in base 6.
= 20 + 21 + 22 + 23
= 3^{2} + 4^{2} + 5^{2} + 6^{2 }(sum
of consecutive squares)
used
as a verb, to "eightysix" means to "ignore" or "get
rid of".

87 
is
the sum of the squares of
the first 4 primes:
2^{2} + 3^{2} + 5^{2} + 7^{2}
= 3
x 29
= 12 + 13 + 14 + 15 + 16 +17
= 44 + 43 = 44^{2}  43^{2}
By tradition, 87 is an unlucky score for cricketers (10013).

88 
is
an hexadecagonal number and the only number known whose square has
no isolated digits. In
fact, its square has all its digits twice: 88^{2} = 7744
is an 'untouchable
number'. An untouchable number is a positive integer that cannot
be expressed as the sum of all the proper divisors of any number.
The sequence of the first untouchable numbers is: 2, 5, 52, 88,
96, 120, 124, 146, 162, 188, 206, 210...
A piano
has 88 keys.
= 23^{2} 
21^{2}
= 13^{2}  9^{2}
= 4^{2} + 6^{2} + 6^{2} (sum
of squares)
= 2^{3} + 2^{3} + 2^{3} + 4^{3} (sum
of cubes)
= 3 + 4 + 5 + ... + 11 + 12 + 13 (sum of consecutive
integers)
= 17 + 19 + 23 + 29 (sum of consecutive primes)
It can
be read the same upside down or when viewed in a mirror.
Eightyeight,
11 letters long, is the longest number that is normally typed
using strictly alternating hands (ignoring the hyphen).
88 is
used in Chinese culture to mean "bye bye"; found in
Chineselanguage chat, text, SMS, IM. 88 is
in fact pronounced in Chinese Mandarin language as "ba ba",
simulating the sound of the English language farewell "bye
bye".
The 8th letter
of the alphabet is 'H'. 8 two times signifies 'HH', shorthand
for the Nazi greeting:"Heil Hitler". 88 is
often found on hate group flyers, in both the greetings and closing
comments of letters written by neoNazis, and in email addresses.

89 
is
a prime and a Fibonacci number.
generates a cyclic number series where each term is the sum of
the square digits of the previous number:
89  145  42  20  4  16  37  58  89  ...
The
decimal expansion of the inverse of
89 (that is 1/89) is just the Fibonacci series, added together
in an appropriate fashion: like a column of decimal fractions,
arranged so the right most digit of the nth Fibonacci
number is in the n+1th decimal place:
.01
.001
.0002
.00003
.000005
.0000008
.00000013
.000000021
.0000000034
.00000000055
.000000000089
.0000000000144
+ ...

.01123595505...
As you
can easily check, 1/89 = 0.01123595595...
89^{2 }=
39^{2} + 80^{2} (Pythagorean
triple)
= 8^{1} +
9^{2}
= 2^{3} + 3^{3} + 3^{3} + 3^{3}

90 
is
a 'perfect
unitary number', that is an integer which is the sum of its
positive proper unitary
divisors (excluding itself).
is the
product of 2 consecutive integers. Such specific numbers are
called 'oblong
numbers' because they correspond to the area of an "oblong" rectangle
with sides n and n+1. Thus, oblong numbers
can also be expressed as n^{2} + n.
is the
number of degrees in a right angle. Moreover, as a compass direction,
90 degrees corresponds to east.
is the
smallest number having 6 representations as a sum of four positive
squares: 90 = 1^{2} + 2^{2} + 2^{2} +
9^{2} = 1^{2} + 2^{2} + 6^{2} +
7^{2} = 1^{2} + 3^{2} + 4^{2} +
8^{2} = 2^{2} + 5^{2} + 5^{2} +
6^{2} = 3^{2} + 3^{2} + 6^{2} +
6^{2} = 3^{2} + 4^{2} + 4^{2} +
7^{2}
= 1
x 2 x 3^{2} x 5
= 2
+ 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 (sum of the
first 9 consecutive even numbers)
= 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24 (2
consecutive sums of consecutive numbers)
= 43 + 47 (sum of 2 successive primes)
= 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive
primes)
= 9^{1} +
9^{2} = 10^{2}  10^{1}
= 0^{1} + 1^{2} + 2^{3} + 3^{4}
= 2^{2} + 3^{2} + 4^{2} + 5^{2} +
6^{2} (sum of consecutive squares)
= 3^{3}  4^{3}  6^{3} + 7^{3}
= 6!
/ 2^{3}
= (2 + 2 + 2)! / (2! x 2! x 2!)
(90^{3} 
1)/(90  1) is a Mersenne
prime, in fact 8191 = 2^{13}  1.
Sturgeon's
Law: "90% of everything is rubbish".
"Success
is 10 percent inspiration and 90 percent perspiration",
Thomas Alva Edison.
Some
'scientific' reviews state humans are "90% water" when
in fact newborns have approx. 78%, 1yearolds around 65%, adult
men about 60%, and adult women around 55%. See mythical
numbers.
"906090" is
considered as an ideal proportion for becoming a model (906090
are the breastwaisthips measurements in centimeters, these
were the alleged measurements of Marilyn Monroe).
Bamboo
can increase its height by 90 centimeters in just one day.
The
Greek and Latin prefixes "enneaconta", "nonaconta",
and "nonaginta" respectively means 'ninety'.
"Nonagerian" is a person who is from 90 to 99 years old.
The compound noun and adjective "fourscore and ten" means
'ninety'.
Even
if you eat a fully organic diet, you would still eat ninety Enumbers (in
fact, our body produces four Enumbers itself). Enumbers are
codes for chemicals used as food additives within the European
Union.
In
Italian, pezzo da novanta (piece of ninety) means 'big
boss' or 'big shot'. In Italy, when someone is or has been frightened
by something unexpected, it is used to say la paura fa novanta! (the
scare does ninety!).
NINETY in
different languages (© G.
Sarcone)
(Old English hund(e)nigontig)

Indoeuropean
 *NEWNKOMT(A) 
Sanskrit  navati 
Greek, Attic  ἐνενήκοντα ENENÈKONTA 
Latin  NONAGINTA  
Italian novanta;
French quatrevingtdix (= 4 x
20 + 10; Swiss French and Belgian nonante);
Spanish and Portuguese noventa;
Catalan noranta; Provençal nonanta;
Romanian nouăzeci; Rumansh navonta;
Sardinian nobànta. 
Old
Germanic  *NIWUNTEHUND  
Dutch negentig;
German neunzig; Danish halvfems (halvfemsindstyve archaic
= half a fifth times twenty); Norvegian nitti;
Swedish nittio; Icelandic níutíu. 
Old
Slavic  DEVETIDESETJ  
Russian девяносто devyanosto;
Czech devadesát; Slovenian devetdeset;
Polish dziewięćdziesiąt. 
Chinese 九十 jiǔshí (complex
form 玖拾). 
Ancient
Egyptian [psḏy.w] pest'ayu;
Akkadian tišā. 
Arabic تسعون tis’un;
Hebrew תשעים teesh'eem;
Maltese: disgħin; Amharic zètèna. 
More
languages
Magyar kilencven.
Mayan lahunhokal (10 of 5 x 20).
Nahuatl nāuhpōhualmahtlāctli.
Suomi yhdeksänkymmentä.
Zulu amashumi ayisishiyagalolunye. 

91 
is
a triangular,
an hexagonal and
also a square pyramidal
number.
is a semiprime and
the smallest pseudoprime to
base 3.
= 1
+ 2 + 3 + ... + 11 + 12 + 13 (sum of consecutive
numbers)
= 1^{2} + 2^{2} + 3^{2} + 4^{2} +
5^{2} + 6^{2} (sum of consecutive
squares)
= 3^{3} +
4^{3}
= 6^{3}  5^{3}
The
traditional dimensions of the tatami
mat (畳) are 91 x 182 cm.
In
2006, 91 per cent of all known executions took
place in China, Iran, Pakistan, Iraq, Sudan and the USA.
Executions in 2006:
China: 1,010+; Iran: 177; Pakistan: 82; Iraq: 65+; Sudan: 65+;
USA: 53; Saudi Arabia: 39+; Yemen: 30+; Vietnam: 14; Kuwait: 10+;
Somalia: 7+; Singapore: 5+; Egypt: 4+; Jordan: 4+; Japan: 4; Malaysia:
4; Indonesia: 3+; Bahrain: 3; Mongolia: 3; North Korea: 2; Syria:
2; Uganda: 2; Bangladesh: 1; Botswana: 1; Equatorial Guinea: 1.
Note: '+' symbol indicates that the figure is a
minimum one; the true figure may be higher due to state secrecy
or a lack of available information.

92 
=
3^{2} + 3^{2} + 5^{2} + 7^{2} (sum
of squares)
is the
number of different arrangements of 8
nonattacking Queens on an 8 by 8 chessboard (i.e. no two
Queens should share the same row, column, or diagonal).
Multiply
92 by 8, then that product by 8, then that product by 8, and so on...
List the products one under the other, shifting the digits two places
to the right as shown below. Continue indefinitely. Add. The sum
converges to a string of 9s:
92
736
5888
47104
376832
3014656
24117248
192937984
9999999...

93 
=
333 in base 5.
With just 9 straight cuts a potato can be divided into 93 pieces.
= 2^{2} +
5^{2} + 8^{2} (sum of squares)
= 13 + 14 + 15 + 16 + 17 + 18

94 
is
a 17gonal number and a Smith
number (a Smith number is a composite number the sum of whose
digits is the sum of the digits of its prime factors excluding
1, here: 9 + 4 = 2 + 4 + 7).
94! 
1 is prime. 
95 
is
the number of planar
partitions of 10.
=
1^{2} + 2^{2} + 3^{2} + 9^{2 }(sum
of squares)
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15
= 5 + 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive
primes)

96 
is
the smallest number that can be written as the difference of
2 squares in
4 ways:
= 25^{2}  23^{2} = 14^{2}  10^{2}
= 11^{2}  5^{2} = 10^{2}  2^{2}
= 2^{5} x
3
=
5! – 4!
= 2^{3} + 2^{3} + 2^{3} + 2^{3} +
4^{3} (sum of cubes)
= 2^{7}  2^{5}

97 
is
a permutable
prime number (the number 97 and
its symmetrical counterpart 79 are
prime numbers).
is a 4dimensional
centered cube prime (prime numbers of the form n^{4}+(n+1)^{4} ).
is the highest 2digit prime number whose cube is zeroless.
97/56
is a good approximation of 3.
curiously,
the numbers 97, 907, 9007, 90007 and 900007
are all prime.
97^{2 }=
65^{2} + 72^{2} (Pythagorean
triple)
97^{3} =
912,673 = (91 + 2 – 6 + 7 + 3)^{3}
= 49^{2} 
48^{2} = 49 + 48
= 2^{3} + 2^{3} + 3^{3} + 3^{3} +
3^{3} (sum
of cubes)
= 2^{4} + 3^{4} (sum
of 4th powers of consecutive primes)
= 29 + 31 + 37 (sum
of consecutive primes)
= 4 x 4! + 4/4 (four
4's diversion)
In the Gregorian
calendar, there are 97 leap days every
4 hundred years.
0.097 is
the average
brain weight of a goldfish (in grams).
97 cm
is the minimum height a visitor must be to ride the Kali
River Rapids ride at Disney World.
97 is
the number of ancient Pyramids existing in Egypt.

98 
is
the lowest number such that its first 5 multiples contain the
digit 9: 098, 196, 294,
392, 490
= 1^{4} +
2^{4} + 3^{4}
=
3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive
primes)
98.14072356
is 9.9066 squared, and it used the digits 1, 2, 3, 4, 5, 6, 7,
8, 9, and 0 exactly once.
'Rachid'
(also 'Rasheed', 'al Rashid', 'ar Rashid', or 'el Rachid'. From
Arabic: راشد; "rightly guided")
is the 98th and next to last name of Allah.
98.6
degrees Fahrenheit is normal body temperature.

99 
is
a palindromic number and a Kaprekar
number.
= 2^{3} +
3^{3} + 4^{3}
= 4 + 5 + 6 + 7 + ... + 11 + 12 + 13 + 14
99/70 is a good approximation of 2.
99^{2} = 9801 and 98 + 01 = 99
2 x 7^{2} +
1 = 99 and 2 x 70^{2} +
1 = 99^{2}
XCIX
or IC?
Rules regarding Roman numerals often state that C cannot be preceded
by I or V, only by X. Thus, one should represent the number 'ninetynine'
as XCIX, not as the 'shortcut' IC.
This 'problem' manifested in questions as to why 1999 was not written
simply IMM or MIM as opposed to the virtually universal MCMXCIX.
Machoumearobilengmonoolemongametsoarobilengmonoolemong means
'99' in the language of the Bassoutos tribe.
Riddle:
What goes "99 bonk"?
Answer: A Centipede with a wooden leg!

100
C

is
the smallest square which
is also the sum of 4 consecutive cubes.
= 99
+ 9/9
= 33 x 3 + 3/3
= 5 x 5 x 5  5 x 5
= (8 + (8 + 8)/8)(8 + (8 + 8)/8)
= 1
+ 2 + 3 + 4 + 5 + 6 + 7 + 8 x 9
= (1 + 2  3  4)(5  6  7  8  9)
= 123 + 45 – 67 + 8 – 9
= 123  45  67 + 89
= 98  76 + 54 + 3 + 21
= ^{2}9
+ ^{3}64
+ 15 + 78
(digits 1 to 9 were used once in the equalities
above)
= 56
+ 34 + 90/(8271) [by G. Sarcone]
= 1/2
+ 38/76 + 49 + 50
= 1/8
+ 63/72 + 49
+ 50
= 3/7
+ 16/28 + 49
+ 50
(the equalities above are pandigital that
is all digits from 0 to 9 were used once)
= 2^{6} + 6^{2}
= 2^{2} x 5^{2}
= 5^{3}  5^{2}
= 1^{3} + 2^{3} + 3^{3} + 4^{3} =
(1 + 2 + 3 + 4)^{2
}= 26^{2}  24^{2 }= 6^{2} + 8^{2}
= 45
+ 55 (sum of consecutive triangular numbers)
= 47 + 53 (sum of consecutive primes)
= 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 (sum of
consecutive primes)
= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 (sum
of consecutive odd number)
Log100
= 2 (base10 logarithm)
Ln100 = 4.60517 (natural logarithm)
2^{100} x 100^{2} +
1 is prime.
The
human brain is a busy place. We each have about 100 billion
neurons. Each of the 100 billion neurons actively communicates
with 1000  10,000 others.
Each of the 100 billion neurons can be 'on' or 'off', and can generate
2^{100,000,000,000} (or roughly 10^{30,000,000,000})
unique brain states. The number of the possible brain states is
greater than the number of particles in the universe!
100
grades (units for angle measures) = 90° = (p/2) radians
100° Celsius
is the boiling point of water.
The 100 billionth
crayon made by Crayola was Periwinkle
Blue.
"A Hundred Thousand
Billion Poems" (French: Cent mille milliards de
poèmes) is a set of 10 sonnets by the French writer
Raymond Queneau that can be combined into 10^{14} different
poems.
Only
one out of one hundred cacao
flowers will become fertilized and grow into a cocoa pod!
Indoeuropean
languages are classified under two major groups: the satam and
the centum group. The terms come from
the words for the number "one hundred" in
representative languages of each group. In the 'centum' group
we find italic (centum), greek (hekaton),
celtic (cent), and germanic languages (hundrad);
in the 'satam' group: indoiranian (satem), baltic (simtas)
and slavic languages (suto).
HUNDRED in
different languages (© G.
Sarcone)
(Old English hund also hundred or hundtêontig)

Indoeuropean
 *(D)KOMTÔM (ten tens) 
Sanskrit  s'atam 
Greek, Attic  ἑκατόν HEKATON 
Latin  CENTUM  
Italian cento;
French, Catalan and Provençal cent;
Spanish cien (ciento);
Portuguese cem (cento);
Romanian o sută (< old
Slavic suto); Rumansh tschien;
Sardinian céntu. 
Old
Germanic  *TEHUNTEHUND also HUN(D)TEHUNTIGUS (ten
tens)  
Dutch honderd (< hundarada =
hundrednumber); German hundert;
Danish hundrede; Norvegian hundre;
Swedish hundra; Icelandic hundrað. 
Old
Slavic  SUTO  
Russian сто sto;
Czech, Slovenian and Polish sto. 
Chinese (一)百 (yī) bǎi (complex
form 壹佰). 
Ancient
Egyptian [š.t] shewt; Akkadian meat. 
Arabic مائة mi'a;
Hebrew מאה me'a;
Maltese: mija; Amharic mèto. 
More
languages
Magyar száz.
Mayan hokal (5 x 20).
Nahuatl mācuīlpōhualli.
Suomi sata.
Zulu ikhulu. 

101 
101^{2} =
20^{2} + 99^{2} (Pythagorean
triple)
= 5!
 4! + 3!  2! + 1!
= 2^{2} + 4^{2} + 9^{2
}= 13 + 17 + 19 + 23 + 29 (sum of consecutive
primes)
= 51^{2}– 50^{2} = 51 + 50
101
x 11 = 111
is the
smallest odd prime that is the sum of a square and its reverse:
100 + 001 = 101
is the
number of partitions of
13.

102 
is
the lowest number with 3 different digits.
=
1^{2} + 4^{2} + 6^{2} + 7^{2} =
2^{2} + 3^{2} + 5^{2} + 8^{2
}(curiously enough 1 + 4 + 6 + 7 = 2 + 3 + 5 + 8)
102^{7} =
12^{7} + 35^{7} + 53^{7} + 58^{7} +
64^{7} + 83^{7} + 85^{7} + 90^{7}
= 2
x 3 x 17
= 33 + 34 + 35 (sum of consecutive numbers)
= 19 + 23 + 29 + 31 (sum of consecutive primes)

103 
is
the only odd prime number that divides the number of bones in
a human body.
= 2^{2} +
5^{2} + 5^{2} + 7^{2}
is
the smallest number other than 1 that is impossible to score
with two darts.

104 
is
a divisor of 25^{2}  1
= 13
x 2^{3} = (13  2^{3})^{3}  13  2^{3}
= 6
+ 8 + 10 + 12 + 14 + 16 + 18 + 20 (sum of consecutive even
numbers)
is the
lowest known number of unit line segments that can exist in the
plane, 4 touching at every vertex.

105 
is
a triangular and a 12gonal number.
is the largest number n known such that n  2^{k} is prime for k=>1
(Erdös
conjecture). For instance: 105  2^{1} = 103; 105 
2^{2} = 101; 105  2^{3} = 97; 105  2^{4} =
89; 105  2^{5} = 73; 105  2^{6} = 41
is the middle of the prime
quadruplet: 101, 103, 107, 109.
=
1 + 2 + 3 + … + 13 + 14 = 6 + 7 + 8 + … + 14 + 15
=
= 12 + 13 + … + 17 + 18 = 15 + 16 + 17 + 18 + 19 + 20 =
= 19 + 20 + 21 + 22 + 23 = 34 + 35 + 36 = 52 + 53 (5
distinct possible sums of consecutive integers)
105^{4} =
22^{4} +
28^{4} +
63^{4} +
72^{4} +
94^{4}
By
70 years, an average person will have lost 105 pounds of
skin particles!

106 
is
the number of trees with
10 vertices.
The sum of the first 106 digits of Pi is prime. 
Sellers
= 25
+ 26 + 27 + 28 (sum of consecutive integers)
= 5^{2} + 9^{2} = 2^{2} + 2^{2} +
7^{2} + 7^{2} 
107 
is
the exponent of a Mersenne
prime.
= 2^{3} +
2^{3} + 3^{3} + 4^{3}
is the
smallest positive integer requiring 6 syllables in English.

108 
is
3 hyperfactorial.
can be written as the sum of a cube and a square (a^{3} + b^{2})
in two ways (3^{3} + 9^{2} and 2^{3} +
10^{2}). This is the smallest number with this property.
Every angle of a pentagon measures 108°.
There are 108 different types of heptominoes.
= 36
x 3 = 27 x 4
= 0^{0} x 1^{1} x 2^{2} x 3^{3}
= 3^{3} + 3^{3} + 3^{3} + 3^{3} =
6^{2} + 6^{2} + 6^{2}
= 12 x 9 and 1 + 0 + 8 = 9
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16
= 3^{2} + 5^{2} + 5^{2} + 7^{2} (sum
of square primes)
= 8 + 15 + 29 + 56 (sum
of consecutive Tetranacci numbers)
is
a significant number in Buddhism. It represents the number of
temptations that human beings face, which tie them to the Wheel
of samsara.
There are many religious elements related to this number  a
bell is rung 108 times to bring in the New Year in Japan.
is
the number of beads in a Buddhist Japa
mala (kind of rosary). They implement the following
formula: 6 x 3 x 2 x 3 = 108
6 senses (sight, sound, smell, taste, touch, thought),
3 aspects of time (past, present, future),
2 conditions of heart (pure or impure),
3 possibilities of sentiment (like, dislike, indifference).
is
the sum of the numbers 4, 8, 15, 16, 23 and 42 that frequently
appear throughout the TV series 'LOST'.
The
diameter of the sun is 108 times the diameter of the Earth.

109 
is
the smallest number which is palindromic in
bases 5 and 9.
Using only the coins in circulation in the UK, you can pay the
sum of 30 pence in 109 different ways.
The interior angle of a tetrahedron is 109° 28’.
Soap bubbles can only join each other at one or two angles: 109
or 120 degrees. No other angles are possible.
= 31
+ 37 + 41 (sum of consecutive primes)
109^{2} = 60^{2} + 91^{2} (Pythagorean
triple)

110 
is
the smallest product of 3 distinct primes of the form 3n 
1:
(3x1  1) x (3x2  1) x (3x4  1) = 2 x 5 x 11 = 110
is the smallest number that is the product of two different substrings.
is the side of the smallest square that can be tiled with distinct
integersided squares (see image below). There are 3 distinct Simple
Perfect Squared Squares with this property. Two 110's with
22 squares were discovered in 1978, one by Duijvestijn using computer
search, the second by Willcocks, who transformed Duijvestijns 110
into a different second 110, and one more 110 with 23 squares was
discovered in 1990 by Duijvestijn. It was Gambini who proved 110
is the minimal square.  Stuart
Anderson
= 5^{2} +
6^{2} + 7^{2} (3 consecutive squares)
= 11^{2}  11^{1} (difference between
powers of the same number)
= 2 x 55 = 5 x 22
= 1x2 + 2x2 + 3x2 +
... + 8x2 + 9x2 + 10x2
110
hertz is the standard frequency of the musical note A or La.
110
is also known as "eleventy" according to the number
naming system invented by J. R. R. Tolkien.
The
expression "to give 110%" means to give a little more
effort than one's maximum effort.

111 
is
the smallest possible magic constant of a 3 x 3 magic
square of distinct primes.
is
the ninth number that stays the same when written upside down.
is also the 'magic constant' of the nQueens
Problem for n = 6.
= 3
x 37
= 12 x 9 + 3
= 16 + 17 + 18 + 19 + 20 + 21
= (1 + 2 + 3 + ... + 34 + 35 + 36) / 6
= 20^{2}  17^{2} = 56^{2}  55^{2} (differences
of squares)
111
x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
Zeno’s
Paradox
The Greek hero Achilles and the turtle are running a foot race.
The tortoise is given a head start of 100 meters. Achilles is catching
up quickly  but the chelonian has moved a further 10 meters during
this time. Achilles quickly pulls up to this short distance as
well. But the turtle has also crawled 1 meter further... In this
way, the turtle always advances a 10th of the distance that Achilles
catches up. Can Achilles ever pass the turtle?
Answer: Yes, he can! After 111 meters (after
111 1/9 meters, to be exact), Achilles will overtake the slowmoving
animal. This paradox is based on a false assumption: it does not
take into account that an infinite progression can have a finite
sum.

112 
is
an abundant
number and a heptagonal
number.
is a number that cannot be written as a sum of 3 squares.
is the maximum number of regions into which 11 circles can divide
a plane.
is the side of the square that
can be tiled with the the fewest
number of distinct integersided squares [see picture
here].  Stuart
Anderson
=
11 + 13 + 17 + 19 + 23 + 29 (sum of consecutive
primes)
= 1x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 (sum of oblong
numbers)
= 2^{4} x 7
= 11^{2}  3^{2} (difference between
two squares)
= 2^{7}  2^{4} (difference
between two powers of the same number)
The
number 112 is used four times in the Bible.
112
is the European emergency phone number.

113 
is
a permutable prime.
355/113 = 3.1415929... A good approximation of .
There
are 13 consecutive divisible integers (nonprimes) between 113
and 127.
For
Tintinologists
113, and not 57b as depicted in the comic book 'the
Calculus Affair' by Hergé, is the number of the street
where the house of the professor Alfredo Topolino is still located
in Nyon. Actually, the current full address is: 113, route de SaintCergue,
1260 Nyon, Switzerland.

114 
appears
in the Padovan
sequence, preceded by the terms 49, 65, 86 (it is the sum
of the first two of these).
= 222
in base 7.
114 = 19 x 6 and 1 + 1 + 4 = 6
The
Muslim holy book the Quran is divided into 114 suras or chapters.

115 
is
a heptagonal
pyramidal number and a lucky
number.
is the number of 8vertex rooted
trees.
= 23
x (2 + 3)
= 5!  5

116 
is
a noncototient.
is a value of n for which n!+1 is prime. 
117 
is
the lowest value of the longest edge in a Heronian
Tetrahedron.
117^{2} +/ 2 are primes.
= 11^{3} 
2^{3} = 5^{3}  2^{3}

118 
is
the smallest even number not differing by 1 or a prime number from
one of its prime neighbors.  S. Wagler
is the lowest number which has 4 different partitions into
3 parts with the same product. 
119 
is
the smallest 3digit number having two prime digit permutations:
191 and 911.
= 17
+ 19 + 23 + 29 + 31 (sum of consecutive primes)

120 
is
a triangular and tetrahedral number.
is the smallest number to appear 6 times in Pascal's
triangle.
= 1
+ 2 + 3 + ... + 13 + 14 + 15 (sum of consecutive
integers)
= 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 (sum of consecutive
triangular numbers)
= 59 + 61 = 23 + 29 + 31 + 37 (sum
of consecutive primes)
= 2^{2} +
4^{2} + 10^{2}
= 3^{1} + 3^{2} + 3^{3} + 3^{4}
= 31^{2}  29^{2} = 17^{2}  13^{2} =
11^{2 }– 1^{2}
= 5!
The truncated
icosidodecahedron (aka great rhombicosidodecahedron)
is an Archimedean solid having 120 vertices,
180 edges, and 62 faces.
120
liters of water are needed to produce 1 glass of
wine (125 ml)!
In India,
the set of Dasavatara
Ganjifa playing cards was composed of 120 circular
cards: that is, ten suits of twelve cards, each suit being based
on one of the ten incarnations of Vishnu.

121 
is
the only square known
of the form 1+p+p^{3}+p^{3}+p^{4},
where p is prime (121
= 3^{0} + 3^{1} + 3^{2} + 3^{3} +
3^{4}).
121
= 38 + 83 (palindromic
sum)
=
5! + 1
= 37 + 41 + 43 (sum of 3 consecutive primes)
= 11^{2}
= 1 + 3 + 5 + 7 + 9 + 11 + ... + 19 + 21 (sum of
consecutive odd numbers)
1/120 =
1/3  1/5  1/8

122 
=
29 + 30 + 31 + 32
= 115 + (1 + 1 + 5)
Palindromic
curiosities with 122:
122 x 122 = 14884 and 48841 = 221 x 221
122 x 213 = 25986 and 68952 = 312 x 221

123 
is
a 42gonal number.
is the 10^{th} Lucas
number.
= ((((4)/.4)^{4!}))))
 4
= 18 + 19 + 20 + 21 + 22 + 23 (sum of consecutive
numbers)
123
x 2^{123} – 1 = 1.30796035 x 10^{39} =
= 1307960347 8523572189 3734614731 5859062783 =
= prime number
123 modulo 12 = 3 
G. Sarcone
Curiosity:
Write down any number (excluding the digit 0):
64861287124425928
Now, count up the number of even and odd digits, and the total
number of digits it contains, as follows:
12  5  17
Then, string those 3 numbers together to make a new number, and
perform the same operation on that:
12517
1  4  5
Keep iterating:
145
1  2  3
You will always arrive at 123.
123
is the symbol of the Holy Trinity: Father, Son and Holy Spirit.
The number 123 is used 3 times in the Bible.

124 
is
an untouchable
number.
is the smallest number such that its first 3 multiples contain
the digit 2.
= 5 +
7 + 11 + 13 + 17 + 19 + 23 + 29 (sum of 8 consecutive primes)
= 2^{7}  2^{2}
= 5^{2} + 5^{2} + 5^{2} + 7^{2} 
125 
is
the smallest number which is both a sum of distinct prime squares
and a prime cube, and is the only number known that contains
all its proper divisors as proper substrings.
= 125 = 5^{1+2}
= 5^{3} = 11^{2} + 2^{2}
Quasquicentennial is
a period of 125 years. Origin: quasqui one and a quarter
(pseudoLatin, appar. QUA(dri) + (se)SQUI ) + CENTENNIAL. 
126 
is
a film size used in easytouse cameras.
= 126 = 6 x 21
= 2 x 3^{2} x 7
= _{9}C_{4}
= 6
+ 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of
consecutive even numbers)
= 2^{7} 
2^{1} (difference
of same powers)
= 4^{2} +
5^{2} +
6^{2} +
7^{2} (sum
of consecutive squares)

127 
is
the 4th Mersenne
prime and an autothecal number.
= 127 = 2^{7}  1 =  1 + 2^{7}
(an autothecal number is an integer n that
can be expressed using all the digits of n once  in order
from left to right  , and the main math operators)
= 1111111_{2}
is in fact the sum of seven powers of 2:
= 2^{0} + 2^{1} + 2^{2 }+ 2^{3} +
2^{4} + 2^{5 }+ 2^{6
}= 2^{7}  2^{0}
= 3
+ 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 (sum of
consecutive primes)
2^{127 }– 1
is the 12th Mersenne
prime number.
You
can arrange the number 127 as a pattern of concentric hexagons.

128 
is
the largest number which cannot be written as the sum of distinct squares.
= 2^{7}
= 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 (sum of consecutive
odd numbers) 
129 
is
the smallest number that can be expressed as the sum of 3 squares in
4 ways. 
130 
is
a noncototient.
is the number of functions from 6 unlabeled points to themselves.
= 1^{2} +
2^{2} + 5^{2} + 10^{2} (sum
of the squares of the first 4 divisors)

131 
is
a permutable
prime, and the smallest palindromic
prime with 3 primes embedded: 13, 3 and 31.
= 41 + 43 + 47 (sum of consecutive primes) 
132 
is
the 6th Catalan
number.
is the smallest number which is the sum of all of the 2digit numbers
that can be formed with its digits. 
133 
is prime,
and its reversal (331) is also prime.
is the smallest integer n for which the sum of the proper
divisors of n divides phi(n).
_{= 2}2^{2}_{+
(2 x 3)}2 _{+ 3}2^{2}_{ 
G. Sarcone}
=
2^{3} +
5^{3}

134 
134^{2} 
67^{2} = 13467
= _{8}C_{1} + _{8}C_{3} + _{8}C_{4} 
135 
=
1^{1} + 3^{2} + 5^{3}
= (1 x 3 x 5)( 1 + 3 +5) 
136 
is
the sum of the cubes of
the digits of the sum of the cubes of
its digits:
136 = 2^{3} + 4^{3} + 4^{3}
1^{3} + 3^{3} + 6^{3} = 244 
137 
divides
11,111,111.
137 and 139 form a twin
prime pair.
it is possible to divide a pizza into 137 tiny pieces with just
17 straight cuts.
is almost the reciprocal of the Sommerfeld's finestructure
constant (137.035999084 ± 0.000000051).
is the maximum number of 7^{th} powers that are needed
to sum to any arbitrarily large number.
is the largest prime factor of 123,456,787,654,321.
is the average
brain weight of a baboon (in grams).
= 4^{2} +
11^{2}
= 2^{7} + 2^{3} + 2^{0}
= 10001 / 73
137^{2} = 88^{2} + 105^{2} (Pythagorean
triple)
137^{2} = 18769,
but 133^{2} = 17689,
281^{2} =78961 and
286^{2} = 81796,
four different squares having the same digits in the solution!

138 
is
the smallest possible product of 3 primes,
one of which is the concatenation of the other two (2 x 3 x 23).
is the sum of four consecutive primes: 29 + 31 + 37 + 41
138
+ 138/2 = 207 and 0.207207207... x 666 =
138
'Untrioctium'
is an unsynthesized chemical element with atomic number 138 and
symbol Uto.
is
considered as an occult number and one of its factors (23) is
also an occult number!

139 
is
the number of unlabeled topologies with
5 elements.
is a prime and a happy number (see also number 86). To find out whether
a number is 'happy' or not, you must square its digits, add them
up, and go on doing this... If you eventually get to 1, you have
a happy number. For instance:
1^{2} + 3^{2} + 9^{2} = 91
9^{2} + 1^{2} = 82
6^{2} + 8^{2} = 100
1^{2} + 0^{2} + 0^{2} = 1 
140 
is
the smallest harmonic
divisor number:
12 / (1/1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/10 + 1/14 + 1/20 + 1/28 +
1/35 + 1/70 + 1/140) = 5
=
1^{2} + 2^{2} + 3^{2} + 4^{2} +
5^{2} + 6^{2} + 7^{2}
140
liters of water are needed to produce 1 cup of
coffee (125 ml)!

141 
is
a Cullen
number (n x 2^{n} + 1 is a Cullen
prime when n = 141).
= 2^{3} +
2^{3} + 5^{3} (sum of cubes)

142 
is
the number of 6vertex planar
graphs. 
143 
is
the smallest quasiCarmichael number
in base 8.
= 11
x 13 (product of twin
primes)
= 1001 / 7
= 2^{2} + 3^{2} + 3^{2} + 11^{2
}=  3^{4}  4^{4}  5^{4 } 6^{4 }+
7^{4
}= 43 + 47 + 53 = 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum
of consecutive primes)

144 
is
the only square in
the Fibonacci
sequence (apart from 1). Moreover, it is the 12th term of
the Fibonacci sequence.
is the
the smallest number whose 5th power is a sum of four 5th powers.
This equality found in 1966 by L. J. Lander and T. R. Parkin
disproved Euler's sum of powers conjecture stating that it always
requires at least n terms of nth power to sum
to a nth power:
144^{5} = 27^{5} + 84^{5} + 110^{5} +
135^{5}
Reversing
12 and 144 in the equality 12^{2} = 144 gives 441
= 21^{2}.
144 =
(1 + 4)!
+ 4! (autothecal
number)
=
71 + 73 (sum of consecutive primes)
= 3! x 4! = 4! + 5!
= 4^{2} + 2 x 8^{2}
= 6^{2} + 3 x 6^{2}
One
of the largest numbers, that is not a power of 10, that has a
specific name: 'gross' (= dozen dozen).

145 
is
a centered
square number and also a pentagonal
number.
is a product of 2 distinct primes: 5 x 29
= 1!
+ 4! + 5!
= 4^{3} + 3^{4}
= 12^{2} + 1^{2} = 8^{2} + 9^{2} (sum
of squares)
= (1^{2} + 2^{2})(2^{2} + 5^{2})
= 3^{3} + 3^{3} + 3^{3} + 4^{3} (sum
of cubes)

146 
is
an octahedral
number. An octahedral number is of the form (2n^{3} + n)/3.
= 222
in base 8
= 5^{2} + 11^{2}
= (1^{2} + 1^{2})(3^{2} + 8^{2})
146^{2} +
1 is prime

147 
is
an odd nonprime number divisible by exactly 3 primes:
= 3 x 7 x 7
is the number of sided 6hexes.

148 
is
an heptagonal
number.
is the number of 6vertex perfect
graphs.
= 3^{2} +
3^{2} + 3^{2} + 11^{2} = 5^{2} +
5^{2} + 7^{2} + 7^{2} (sum
of squares)
= 2^{2} + 12^{2}
= (1^{2} + 1^{2})(5^{2} + 7^{2}) 
149 
is
the concatenation of the first 3 positive squares.
= 6^{2} +
7^{2} + 8^{2} (sum of consecutive
squares)
= 2^{3} + 2^{3} + 2^{3} + 5^{3} (sum
of cubes)
= 23 + 43 + 83 (sum of 3 primes which are not sum
of two squares)

150
CL 
is
the smallest n for which n + n times
the n^{th} prime is square.
has
no prime factors larger than 5.
is the
longest interval between consecutive twin
prime pairs less than a thousand. It occurs between [659,
661] and [809, 811].
2^{150 }– 3
and 2^{150} – 5 are twin
primes.
sin(150°)
= cos(30°) = cos(60°) = cos(300°) = 0.5
cos(150°) = sin(240°) = sin(300°)
= cos(210°) = –3^{1/2}/2
= 7
+ 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum of consecutive
primes)
Dunbar's
number is a theoretical value with implications
in sociology and anthropology. It is actually a theoretical
cognitive limit to the number of people with whom one can
maintain stable social relationships. No precise value has
been proposed for Dunbar's number, but a commonly cited approximation
is 150.
Sesquicentennial is
a period of 150 years.

151 
is
a palindromic
prime.
is the total number of types of Pokémon in the original
set.
149 and 151 form a pair of twin
primes.
= 2^{2} +
7^{2} + 7^{2} + 7^{2} (sum
of prime squares)

152 
=
2^{3} x 19
is the largest known even number that can be expressed as the sum
of 2 primes in exactly 4 distinct ways:
3 + 149, 13 +139, 43 + 109, 73 + 79
is the smallest integer that can be represented as the sum of the
cubes of two distinct odd primes: 152 = 3^{3} + 5^{3}
= 31 + 37 + 41 + 43 (sum of consecutive primes)
= 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 
153 
is
a triangular number:
= 1 + 2 + 3 + 4 + 5 + ... + 15 + 16 + 17
is also an hexagonal number and and one of six known truncated
triangular numbers.
= 3^{2} x 17
= 1^{3} + 5^{3} + 3^{3} (is
the smallest number that is the sum of powers of its own digits,
where the power is the same as the number of digits. The next few
numbers with this property are: 370, 371, 407, 1634...)
= 1! + 2! + 3! + 4! + 5!
= 3^{2} + 12^{2}
= 77^{2}  76^{2}
153 = 3 x
51
153 / 999 = 0.153153153... (cyclic
number)
The
sum of digits of 153 is a perfect square:
1 + 5 + 3 = 9 = 3^{2}
The sum of aliquot divisors of 153 is also a perfect square:
1 + 3 + 9 + 17 + 51 = 81 = 9^{2}
The
integer 153 is the only three digit number (in the decimal
system) that has the property: abc = a(b!) + b(c!) + c(a!)
153 = 1x5!+ 5x3! + 3x1!  posted by Amit Sharma
from Oklahoma State University
Phil
Kohn discovered an interesting property of 153:
Start with any integer that is a multiple of 3. Sum the cubes
of its digits to obtain a second number. Then keep repeating
this procedure. In a finite number of steps you will arrive at
the dead end of 153. Let's try with the number
162:
1^{3} + 6^{3} + 2^{3} = 225;
2^{3} + 2^{3} + 5^{3} = 141;
1^{3} + 4^{3} + 1^{3} =66;
6^{3} + 6^{3} = 432;
4^{3} + 3^{3} + 2^{3} = 99;
9^{3} + 9^{3} = 1458;
1^{3} + 4^{3} + 5^{3} + 8^{3} =
702;
7^{3} + 2^{3} = 351;
et voila
3^{3} + 5^{3} + 1^{3} = 153
On
adding the number 153 to its reverse we obtain 504, whose square
is the smallest square which can be expressed as the product
of two different numbers which are reverse of one another:
153 + 351 = 504, and 504^{2} = 288 x 882
153 + 315
+ 531 = 351 + 135
+ 513
"Ascendit
Simon Petrus et traxit rete in terram plenum magnis piscibus, centum
quinquaginta trium". (Iohannem, 21:11)
The net which Simon drew out of the sea at Tiberias contained 153 fishes (Holy
Bible, John 21:11).

154 
is
a nonagonal number
and the smallest integer which is palindromic in
bases 6, 8, and 9.
= 2 x
7 x 11
= 2^{2} + 2^{2} + 5^{2} + 11^{2} (sum
of squares) 
155 
is
the sum of the primes between
its smallest and largest prime
factor.
= 3^{3} +
4^{3} + 4^{3} (sum of cubes)
= 5^{1} + 5^{2} + 5^{3}
= 2^{2} + 3! + 5! + 7^{2}  11  13 
156 
is
the number of 6vertex graphs.
is the number of strikes a clock will strike in the course of a
day.
The equalities below are written with all digits from 1 to 9 once:
2 x 78 = 156 = 39 x 4
1/156 =
1/3  1/4  1/13

157 
is
a prime number and the largest known number whose square contains
the same digits as its successor:
157^{2} = 24649 and 158^{2} = 24964
CIVIL is
the anagram of the Roman numeral CLVII (157).

158 
is
the number of planar
partitions of 11.
= 3
+ 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum
of consecutive odd numbers)

159 
is
the number of isomers of C_{11}H_{24}.
=
47 + 53 + 59 (sum of consecutive primes)
= 2^{2} + 3^{2} + 5^{2} + 11^{2} (sum
of squares)
is the
lowest number that it is impossible to finish on with 3 darts. 
160 
is
the number of 9iamonds.
= 2^{3} +
3^{3} + 5^{3} (sum of cubes)
= 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum
of consecutive primes) 
161 
is
a hexagonal
pyramidal number.
When 161 is rotated 180 degrees it becomes a palindromic prime. 
162 
=
2 x 3^{4}
≈ 100 x Golden Number 
163 
is
a lucky
prime and the largest Heegner
Number.
≈ 69/(π 
e)
≈ 2^{5}·log_{e}163
=
2^{3} + 3^{3} + 4^{3} + 4^{3} (sum
of cubes)
= 1 + 2 x 3^{4}

164 
=
2^{2} x 41
= 3^{2} +
3^{2} + 5^{2} + 11^{2} (sum
of squares) 
165 
= _{11}C_{3 }= _{11}C_{8}
= (9
x 10 x 11) / 6 
166 
is
a centered triangular number.
is the number of monotone Boolean
functions of 4 variables.
is the smallest Smith
number that is prime when reversed (661), turned upsidedown
(991), and reversedthenturnedupsidedown (199).
166!1
is a factorial minus one prime.
Mathematically
correct result derived by incorrect calculation method (anomalous
cancellation):
166 = 166 = 1
664 664 4 
167 
is
a prime number.
167 x 701 = 117067
(the product has the same digits as its factors)
In
backgammon, starting from the initial position, each player has
a pip count of 167.

168 
n^{7}  n is
divisible by 168 when n is odd.
= 37
+ 41 + 43 + 47 (sum of consecutive primes)
There
are 168 hours in a week.
Many
Chinese products bear price tags with the number 168 on them,
because 168 sounds like the Chinese expression, "lucky all
the way".

169 
is
a square whose
digits are nondecreasing.
169
= 13^{2} and in reverse order 31^{2} = 961
= 13 + 17 + 19 + 23 + 29 + 31 + 37 (sum of consecutive
primes)
= 5^{2} + 12^{2}
= 8^{3} – 7^{3} (difference
of consecutive cubes)
= ( 3^{2} + 2^{2})(3^{2} + 2^{2})
= (16 + 9) + (16 x 9)
169^{2} =
119^{2 }+ 120^{2} (Pythagorean
triple)
169
= 19  6

170 
is
the smallest number n for which phi(n) and sigma(n) are
both square. 
171 
is
a palindromic and triangular
number.
=
2^{3} + 2^{3} + 3^{3} + 4^{3} +
4^{3} (sum of cubes)
10^{171} 
171 is prime.  Broadhurst & Titanix 
172 
=
444 in base 6.
= 1 x 2^{2} x 43
is the sum of 2 consecutive primes: 83 + 89
= 3^{3} + 3^{3} + 3^{3} + 3^{3} +
4^{3} (sum of cubes) 
173 
is
a prime number.
is the sum of 3 consecutive primes: 53 + 59 + 61
=
2^{2} + 13^{2}
To predict
the eclipses, the ancient Siberians used the strings of dots:
{53+33}+{30}+{23}+{16}+{7+9+1+1} which are equal to 173, and
the periods of time (24 hours)  86, 30, 23, 16, 7, 9, 1, 1
 are noteworthy for the calculations.
It takes around 173 years for all the water in Lake Superior to
be changed.

174 
is
a 59gonal number
= 2 x 3 x 29
= 5^{2} + 6^{2} + 7^{2} + 8^{2} 
175 
is
the smallest number n>1 such that n^{6}+6
is prime.
= 1^{1} + 7^{2} + 5^{3}
Demisemiseptcentennial or quartoseptcentennial is
a period of 175 years. 
176 
is happy
number and an octagonal
and pentagonal number.
is the number of possible partitions of the number 15.
= 2^{3} +
2^{3} + 2^{3} + 3^{3} + 5^{3} (sum
of cubes)

177 
is
a 60gonal number.
is the number of 7edge graphs.
is the 'magic constant' of the Chen
prime magic square:
Chen's
Magic Square
17 
89 
71 
113 
59 
5 
47 
29 
101 
=
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 +
43 + 47 (sum of the first 15 consecutive primes)

178 
is
a 31gonal number.
Strangely
enough, 178 and 196 are related... In fact, 178 has a square with
the same digits as 196:
178^{2} = 31,684
196^{2} = 38,416
178 has also a cube with
the same digits as 196:
178^{3} = 5,639,752
196^{3} = 7,529,536
=
2^{2} + 2^{2} + 7^{2} + 11^{2} (sum
of consecutive square primes)

179 
is
a prime number.
3^{2} +
691^{2 }+ 2293^{2 }= 179^{3}

180 
is
the total number of degrees in a triangle.
arctan1
+ arctan2 + arctan3 = 180

181 
is
a strobogrammatic prime.
It can be read the same upside down or when viewed in a mirror.
181^{2} =
105^{3}  104^{3} (solution of x^{2} = y^{3}  z^{3})

182 
is
a palindromic and undulating number, if written in the ternary,
the negaternary, or the nonary numeral
systems.
= 3^{3} +
3^{3} + 4^{3} + 43 (sum of cubes)
= 14^{2}  14

183 
is
a central polygonal number, n^{2}  n +
1, where n = 14.
is a number that cannot be written as a sum of 3 squares.
is the smallest integer n so that n concatenated
with n+1 is square: 183184 =
428^{2}
is
111 in base 13.
=
32^{2}  29^{2} = 92^{2}  91^{2} (difference
of squares)
= 2^{2} + 3^{2} + 7^{2} + 11^{2} (sum
of 4 distinct square primes)
λοπαδοτεμαχοσελαχογαλεοκρανιολειψανοδριμυποτριμματο
σιλφιοκαραβομελιτοκατακεχυμενοκιχλεπικοσσυφοφαττο
περιστεραλεκτρυονοπτοκεφαλλιοκιγκλοπελειολαγῳοσιραιο
βαφητραγανοπτερύγων transliterated
as: Lopadotemachoselachogaleokranioleipsanodrimhypo
trimmatosilphioparaomelitokatakechymenokichlepi
kossyphophattoperisteralektryonoptekephalliokigklo
peleiolagoiosiraiobaphetraganopterygon is a 183letter Greek
word coined by Aristophanes in Ἐκκλησιάζουσαι (Assemblywomen);
and it represents a fictional dish created with the ingredients
which make up its name (all kinds of dainties: sauces, fish, poultry
and other meat). It is also cited as the longest ancient Greek
word ever written.
the 183th day
of the year (July 2) is the midpoint of a nonleap year (because
there are exactly 182 days before and 182 days after).

184 
is
a Kaprekar
constant in base 3.
is the largest number of regions into which a plane can be divided
by 14 circles.
1 /
184 = 0.0054347826086956521739130...
(the reciprocal of 184 contains a 22digit long repeating numeral
pattern)
= 181
+ 3 = 179 + 5 = 173 + 11 = 167 + 17 = 137 + 47 = 131 + 53 = 113
+ 71 = 101 + 83 (sum of 8 distinct pair of primes:
ref. Godlbach conjecture)
= 2^{3} x 23 (concatenation
of the first 2 primes)
= 41 + 43 + 47 + 53 (sum of consecutive primes)
= 25^{2}  21^{2} = 47^{2}  45^{2} (difference
of squares)

185 
=
13^{2} + 4^{2} = 11^{2} + 8^{2}
= 5 x 37
1/185 = 0.0054054054054...
185 x 3/5 = 111 
186 
is
a 14gonal number and a 63gonal number.
= 89
+ 97 (sum of consecutive primes)
= 2^{2} + 2^{2} + 3^{2} + 13^{2 }(sum
of squares)

187 
is
the lowest quasiCarmichael number in base 7.
= 59
+ 61 + 67 (sum of 3 consecutive primes)
= 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 (sum
of 9 consecutive primes)
= 2^{3} + 3^{3} + 3^{3} + 5^{3} (sum
of cubes)
The
189th century would be the first to contain as few as five prime
years: 18803, 18839, 18859, 18869 and 18899...

188 
is
the number of semigroups of
order 4.
= 2^{2} +
3^{2} + 7^{2} + 11^{2} (sum
of squares)

189 
is
the number of digits we need to write down all numbers from 1
to 99. The general formula to find out how many digits are needed
to write down all numbers from 1 to N is:
n(N + 1)  (10n  1)/9 [where n is
the number of digits in N]
is a Kaprekar
constant in base 2.
is a
divisor of 55^{2}  1
A "prime
quadruplet" is a set of 4 primes of the form p, p+2, p+6,
and p+8. Interestingly, the product of 4 primes in
a prime quadruplet (except for 5, 7, 11, 13) always ends in 189.
For instance: 11 x 13 x 17 x 19 = 46,189
= 4^{3} +
5^{3} = ( 3)^{3} + 6^{3} (sum
of cubes)
=
24 + 25 + 26 + 27 + 28 + 29 + 30 (sum of consecutive
numbers)

190 
is
is a triangular number, a hexagonal number.
is the largest number with the property that it and its distinct prime factors
are palindromic in Roman
numerals:
190 = CXC = II · V · XIX
 Thoms
=
4^{2} + 5^{2} + 6^{2} + 7^{2} +
8^{2 }(sum of consecutive squares)
=
1 + 2 + 3 + ... + 17 + 18 + 19 (sum of consecutive
numbers)

191 
is
a palindromic
prime.
By adding
up the values of the common US coins, we obtain 191 cents (silver
dollar + half dollar + quarter + dime + nickle + penny).

192 
is
the smallest number with 14 divisors.
= 5
+ 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 (sum
of consecutive primes)
= 2^{8}  2^{6} =
4^{4}  4^{3}

193 
is
a twin prime with 191, and the only known odd prime p for
which 2 is not a primitive
root of 4p^{2}+1.
A
scalene triangle having sides 193, 194 and 195
(consecutive integers) has an area that is also an integer (16296
units^{2}). Such triangles whose
sidelengths and area are all integers are called Heronian
triangles.
=
(9^{3}  7^{3}) / 2

194 
is
the smallest number that can be written as the sum of 3 squares in
5 different ways.
is the
product of two distinct primes: 2 x 97
= 7^{2} +
8^{2} + 9^{2} (sum of consecutive
squares) 
195 
is
the smallest number that can be expressed as a sum of distinct
squares in 16 different ways.
is the lowest value of n such that _{2n}C_{n} is
divisible by n^{2}. 
196 
is
a squarefull number. A number is said to be squarefull if for
every prime, p, that divides it, p^{2} also
divides it: 196 = 2^{2} x 7^{2}
is the
lowest Lychrel number, that is the lowest number that is not
known to reach a palindrome when
repeatedly added to its reverse. For example, let's begin with
the number 392... If we reverse its digits and add it to the
original number, we get 685. Like so:
392 + 293 = 685
Since this new number is not a palindrome, we can continue the
reverseandadd process and it may become one:
685 + 586 = 1,271
1,271 + 1,721 = 2,992 is a palindrome.
Most of the time only a few iterations (less than 10) are needed
before the number becomes a palindrome. In the case of 196, it
is believed that it will never become a palindrome, no matter how
many iterations are performed on the number. But is 196 really
immune to producing palindromes? No one’s yet offered a conclusive
proof.
= 1
+ 3 + 5 + … + 23 + 25 + 27 (sum of consecutive
odd numbers)
196 =
14^{2}
169 =
13^{2}
961
= 31^{2}
196
= (7
 i7)^{4} =
(7
+ i7)^{4 }(power
of complex numbers)

197 
is
the only 3digit Keith
number.
= 2^{3} +
4^{3} + 5^{3} (sum of cubes)
= 99^{2}  98^{2}

198 
is
a 67gonal number and a practical
number.
is between the twin
prime pair of 197 and 199.
abc  cba =
198 (replace each letter with a digit so that a=n, b=n1
and c=n2. E.g. 765  567 = 198)
=
11 + 99 + 88
= ( 100 – 1) ( 100 – 98)
=
55% of 360
The
IKEA catalog has a print run of 198 million copies
in 27 languages and distributed in 35 countries (2009).

199 
is
an 'invertible prime' (turned upside down, a different prime
is formed: 661).
is also an 'absolute' or 'permutable
prime', meaning that 919 and 991 are primes as well!
is the 11^{th} Lucas
number.
Adding
210 to 199 over and over again, you get 8 more
primes that can be arranged together into a 3x3 magic
square:
1669 
199 
1249 
619 
1039 
1459 
829 
1879 
409 
=
2^{2} + 5^{2} + 7^{2} + 11^{2} (sum
of consecutive square primes)
= 3^{2} + 4^{2} + 5^{2} + 6^{2} +
7^{2} + 8^{2 }(sum
of consecutive squares)
=
61 + 67 + 71 (sum of 3 consecutive primes)
= 31 + 37 + 41 + 43 + 47 (sum of 5 consecutive primes)
Mathematically
correct result derived by incorrect calculation method (anomalous
cancellation):
199 = 199 = 1
995 995 5 
200 
appears
in the Padovan
sequence.
=
66^{3 }– 536^{2}
=
2^{3} x 5^{3}
=
7^{0} + 7^{1} + 7^{2} + 7^{3}
=
2^{3} + 4^{3} + 4^{3} + 4^{3} (sum
of cubes)
11113^{2 }– 200^{2} =
11313 x 10913 = 123,458,769
and the reverse of the digits of all of the above numbers (200 excepted)
have the same property:
31111^{2} – 200^{2} = 31311
x 30911 = 967,854,321
The
number 200, according to Bullinger's study of biblical literature,
signifies 'insufficiency'.
The
word 200 (ducenti) has a double meaning in Latin: "two hundred" and/or "to
the leading man" (dative case of 'ducens', leading). 
G. Sarcone
200
liters of water are needed to produce 1 glass of
milk (200 ml)!
Bicentennial is
a period of 200 years.
