Zoo of Numbers: Numbers 70 to 200

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What's Special About This Number?

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

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is 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(6²- 1²) = 2(6 + 1)(6 -1)

70² = 1² + 2² + 3² + 4² + ... + 22² + 23² + 24²

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 house-God, 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)

Indo-european | *SEPT'MKOMT(A) | Sanskrit | saptati | Greek, Attic | ἑβδομήκοντα HEBDOMEKONTA | Latin | SEPTUAGINTA |

Italian settanta; French soixante-dix (= 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.

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)
= 36² - 35² = 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)

71² = 7! + 1
71² = 2⁷ + 17³ (sum of prime powers of two prime numbers)
71³ = 357911 (consecutive odd numbers)
71⁴ = 136³ + 4785²

"Brown numbers" are pairs of integers m and n satisfying:
m² = n! + 1 (Brocard's problem)
There are only 3 known pairs: (5, 4), (11, 5), (71, 7)

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
= 2³(2³ + 1) = 3²(3² - 1)
= 2³ · 3²
= 11² - 7² = 19² - 17² (difference of 2 consecutive squared primes)
= 2³ + 4³ = 3⁴ - 3²

72 = 24 x 3 and reversely 27 = 24 + 3

72⁵ = 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵

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]

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 10⁴ + 1 = 10001 (composite Fermat number).

Every positive integer can be expressed as a sum of 73 or fewer sixth powers.

Sheldon Primes:
The mirror of 73 - the 21st prime number - is 37 - the 12th prime number. Do other similar pairs exist?

The sum of the first 73 odd primes is divisible by 73.

73 = 37² – 36² = 37 + 36

73² = 48² + 55² (Pythagorean triple).

365 days in a year : 5 = 73

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 = 6² + 6²
73 = 3² + 8²
74 = 5² + 7²
- 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.

is a nontotient.
is the number of non-Hamiltonian polyhedra with a minimum number of vertices.

= 17 + 18 + 19 + 20

74% is the storage density of a volume filled with small spheres.

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!

75⁵ = 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵

75 liters of water are needed to produce 1 glass of beer (250 ml)!

A dollar bill contains 75% cotton and 25% linen.

is an automorphic number:
76² = 5776
76³ = 438976
76⁴ = 33362176
76⁵ = 2535525376...

= 8 + 13 + 21 + 34 (sum of consecutive Fibonacci numbers)
= 3² + 3² + 3² + 7² (sum of square primes)

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:
4² + 5² + 6²
is the sum of the first 8 prime numbers:
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19

= 1001 / (77 - 37 - 27) - By G. Sarcone
= 39² - 38²
= 9² - 2²

77² = 18² + 19² + 20² + … + 26² + 27² + 28² (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)

Seventy-seven is the smallest number requiring five syllables in English.

French speaking people have two different names for seventy-seven: 'Soixante-dix-sept' (sixty and seventeen) and 'septante-sept' (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.

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)

78³ = 39³ + 52³ + 65³

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 "seventy-eights")
as they played at 78 revolutions per minute.

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, ...)

= 40² - 39² = 40 + 39

= 2⁷ - 7²
= 2² + 5² + 5² + 5² (sum of squares)

≈ 8.888²

x² - 79x + 1601, if you replace x in the previous expression with any number from 1 to 79, you always obtain a prime number!

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^(n-1) = 5 x 2⁴

= 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)
= 2³ + 2³ + 4³ (sum of 3 positive cubes)

= 4² + 8² = 4² + 4³
= 9² - 1² = 292² - 44³

= 6! / 3²

80² = 4³ + 8³ + 12³ + 16³

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:
'quatre-vingts', 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 quatre-vingts jours) is a classic adventure novel by the French writer Jules Verne.

村八分 (mura hachibu) is a term in Japanese culture that describes the act of socially shunning someone. Originating in the Edo Era, it involves excluding a villager from 80% (hachibu) of village (mura) life as a consequence for violating social norms, such as neglecting communal work. The excluded aspects encompass significant events like births, weddings, and construction. However, the transgressor is still involved in communal matters essential for the well-being of the village, such as firefighting and managing funerals.

EIGHTY in different languages (© G. Sarcone) (Old English hundeahtatig)

Indo-european | *OKTÔKOMT(A) | Sanskrit | aśîti | Greek, Attic | ὀγδοήκοντα OGDOKONTA | Latin | OCTOGINTA | ; Medieval Latin | OCTUAGINTA |

Italian ottanta; French quatre-vingts (= 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.

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)

= 3⁴ = 9²
= 2⁵ + 7²
= 41² - 40² = 41 + 40
= 1² + 4² + 8²

= (1 + 2 + 2 + 4)² = 1³ + 2³ + 2³ + 4³

x² - 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 0-9 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 n⁴. For instance:
(1) + (4 + 5 + 6) + (11 + 12 + 13 + 14 + 15) = 3⁴ = 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 'sex-N-beer', 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).

is an happy number, and a companion Pell number.
is a 15-gonal number and a centered 27-gonal 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 6-hexes (with 82 6-hexes it is possible to form an hexagonal ring).

= 1 + 5 + 21 + 55 (sum of Fibonacci numbers)
= 19 + 20 + 21 + 22 (sum of consecutive integers)
= 1² + 9²
= 2² + 2² + 5² + 7² (sum of square primes)
= 2(4² + 5²)

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)

Start with 82, then count backwards to 1, and you magically obtain a prime:
8281807978777675747372717069686
7666564636261605958575655545352
5150494847464544434241403938373
6353433323130292827262524232221
2019181716151413121110987654321

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 'eighty-two'.
The noun and adjective "duooctogintennary" means 'a group of 82' or 'a 82-year 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.

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 10⁸³ elementary particles with non-zero mass in the known universe.

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

= 2² + 4² + 8² = 4¹ + 4² + 4³
= 2⁵ + 3³ + 5² (is the smallest number that can be expressed as the sum of 3 distinct primes raised to distinct prime exponents)
= 3² + 5² + 5² + 5² (sum of cubes)

N = n⁷ - n, N is divisible by 42, 84, 168 when n is odd.

1000 + 84 = 1084 (one thousand eighty-four) 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 Eighty-Four in Pennsylvania, USA.

is the largest known integer n for which
1² + 2² + 3² + 4² + ... + (n-1)² + n² = 1 + 2 + 3 + 4 + ... + (q-1) + q has a solution. In fact:
1² + 2² + 3² + 4² + ... + 85² = 1 + 2 + 3 + 4 + ... + 645 = 208,335

is a centered triangular number
.

85/38 is a good approximation of 5.

= 43² – 42² = 43 + 42
= 2² + 9²
= 6² + 7² (sum of 2 consecutive squares)
= 4⁰ + 4¹ + 4² + 4³ (sum of powers of 4)

85² = 13² + 84² = 36² + 77² (Pythagorean triples)

(85¹¹ - 85)/11 - 1 and 85¹¹ - 85)/11 + 1 are twin primes

is an happy number.
is the largest known n for which 2ⁿ contain NO zeros:
2⁸⁶ = 77,371,252,455,336,267,181,195,264 (26 digits).

= 222 in base 6.
= 20 + 21 + 22 + 23
= 3² + 4² + 5² + 6² (sum of consecutive squares)

used as a verb, to "eighty-six" means to "ignore" or "get rid of".

is the sum of the squares of the first 4 primes:
2² + 3² + 5² + 7²

= 3 x 29
= 12 + 13 + 14 + 15 + 16 +17
= 44 + 43 = 44² - 43²

By tradition, 87 is an unlucky score for cricketers (100-13).

is an hexadecagonal number and the only number known whose square has no isolated digits. In fact, its square has all its digits twice: 88² = 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.

= 23² - 21²
= 13² - 9²
= 4² + 6² + 6² (sum of squares)
= 2³ + 2³ + 2³ + 4³ (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.

Eighty-eight, 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 Chinese-language 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 neo-Nazis, and in e-mail addresses.

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...

89² = 39² + 80² (Pythagorean triple)

= 8¹ + 9²
= 2³ + 3³ + 3³ + 3³

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 n² + 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 = 1² + 2² + 2² + 9² = 1² + 2² + 6² + 7² = 1² + 3² + 4² + 8² = 2² + 5² + 5² + 6² = 3² + 3² + 6² + 6² = 3² + 4² + 4² + 7²

= 1 x 2 x 3² 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)

= 9¹ + 9² = 10² - 10¹
= 0¹ + 1² + 2³ + 3⁴
= 2² + 3² + 4² + 5² + 6² (sum of consecutive squares)
= 3³ - 4³ - 6³ + 7³

= 6! / 2³
= (2 + 2 + 2)! / (2! x 2! x 2!)

(90³ - 1)/(90 - 1) is a Mersenne prime, in fact 8191 = 2¹³ - 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%, 1-year-olds around 65%, adult men about 60%, and adult women around 55%. See mythical numbers.

"90-60-90" is considered as an ideal proportion for becoming a model (90-60-90 are the breast-waist-hips 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 E-numbers (in fact, our body produces four E-numbers itself). E-numbers 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)

Indo-european | *NEWNKOMT(A) | Sanskrit | navati | Greek, Attic | ἐνενήκοντα ENENÈKONTA | Latin | NONAGINTA |

Italian novanta; French quatre-vingt-dix (= 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.

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)
= 1² + 2² + 3² + 4² + 5² + 6² (sum of consecutive squares)

= 3³ + 4³
= 6³ - 5³

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.

= 3² + 3² + 5² + 7² (sum of squares)
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 (sum of consecutive numbers)

is the number of different arrangements of 8 non-attacking Queens on an 8 by 8 chessboard (i.e. no two Queens should share the same row, column, or diagonal).

		92
		 736
		  5888
		   47104
		    376832
		     3014656
		      24117248
		       192937984
		9999999...

= 333 in base 5.
With just 9 straight cuts a potato can be divided into 93 pieces.

= 2² + 5² + 8² (sum of squares)
= 13 + 14 + 15 + 16 + 17 + 18

is a 17-gonal 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).

is the number of planar partitions of 10.
is a number that cannot be written as a sum of 3 squares.

= 1² + 2² + 3² + 9² (sum of squares)
= 5 + 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive primes)

is the smallest number that can be written as the difference of 2 squares in 4 ways:
= 25² - 23² = 14² - 10²
= 11² - 5² = 10² - 2²

= 2⁵ x 3
= 5! – 4!
= 2³ + 2³ + 2³ + 2³ + 4³ (sum of cubes)
= 2⁷ - 2⁵

is a permutable prime number (the number 97 and its symmetrical counterpart 79 are prime numbers).
is a 4-dimensional centered cube prime (prime numbers of the form n⁴+(n+1)⁴ ).
is the highest 2-digit 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.

97² = 65² + 72² (Pythagorean triple)
97³ = 912,673 = (91 + 2 – 6 + 7 + 3)³

= 49² - 48² = 49 + 48
= 2³ + 2³ + 3³ + 3³ + 3³ (sum of cubes)
= 2⁴ + 3⁴ (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.

is the lowest number such that its first 5 multiples contain the digit 9: 098, 196, 294, 392, 490

= 1⁴ + 2⁴ + 3⁴

= 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.

is a palindromic number and a Kaprekar number.

= 2³ + 3³ + 4³
= 4 + 5 + 6 + 7 + ... + 11 + 12 + 13 + 14
99/70 is a good approximation of 2.
99² = 9801 and 98 + 01 = 99

2 x 7² + 1 = 99 and 2 x 70² + 1 = 99²

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 'ninety-nine' 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!

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
= ²9 + ³64 + 15 + 78
(digits 1 to 9 were used once in the equalities above)

= 56 + 34 + 90/(82-71) [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)

= 2⁶ + 6²
= 2² x 5²
= 5³ - 5²
= 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)²
= 26² - 24² = 6² + 8²

= 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 (base-10 logarithm)
Ln100 = 4.60517 (natural logarithm)

2¹⁰⁰ x 100² + 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 10¹⁴ different poems.

Only one out of one hundred cacao flowers will become fertilized and grow into a cocoa pod!

Indo-european 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 (he-katon), celtic (cent), and germanic languages (hund-rad); in the 'satam' group: indo-iranian (satem), baltic (simtas) and slavic languages (suto).

The word hundred is derived from Old Norse “hundrath” which actually meant 'long hundred’, that is 120 and not 100.

HUNDRED in different languages (© G. Sarcone) (Old English hund also hundred or hundtêontig)

Indo-european | *(D)KOMTÔM (ten tens) | Sanskrit | s'atam | Greek, Attic | ἑκατόν HE-KATON | 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 (< hunda-rada = hundred-number); 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.

101² = 20² + 99² (Pythagorean triple)

= 5! - 4! + 3! - 2! + 1!
= 2² + 4² + 9²
= 13 + 17 + 19 + 23 + 29 (sum of consecutive primes)
= 51²– 50² = 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.

is the lowest number with 3 different digits.

= 1² + 4² + 6² + 7² = 2² + 3² + 5² + 8²
(curiously enough 1 + 4 + 6 + 7 = 2 + 3 + 5 + 8)

102⁷ = 12⁷ + 35⁷ + 53⁷ + 58⁷ + 64⁷ + 83⁷ + 85⁷ + 90⁷

= 2 x 3 x 17
= 33 + 34 + 35 (sum of consecutive numbers)
= 19 + 23 + 29 + 31 (sum of consecutive primes)

is the only odd prime number that divides the number of bones in a human body.

101 and 103 form a twin prime pair.

= 2² + 5² + 5² + 7²

is the smallest number other than 1 that is impossible to score with two darts.

If you fold a paper in half 103 times it will be larger than the observable Universe: 93 billion light-years! (Theoretically). But how can a 0.1-millimeter-thick paper get to be as thick as the Universe? The answer is quite simple: exponential growth… The paper when folded doubles, then quadruples, then increases in thickness by leaps and bounds, which means eight and then 16 times and then 32 times, and so on. More information here.

is a divisor of 25² - 1

= 13 x 2³ = (13 - 2³)³ - 13 - 2³

= 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.

is a triangular and a 12-gonal number.
is the largest number n known such that n - 2^k is prime for k=>1 (Erdös conjecture). For instance: 105 - 2¹ = 103; 105 - 2² = 101; 105 - 2³ = 97; 105 - 2⁴ = 89; 105 - 2⁵ = 73; 105 - 2⁶ = 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)
105⁴ = 22⁴ + 28⁴ + 63⁴ + 72⁴ + 94⁴

By 70 years, an average person will have lost 105 pounds of skin particles!

is the number of trees with 10 vertices.
The sum of the first 106 digits of Pi is prime. - Sellers

is the exponent of a Mersenne prime.

= 2³ + 2³ + 3³ + 4³

is the smallest positive integer requiring 6 syllables in English.

is 3 hyperfactorial.
can be written as the sum of a cube and a square (a³ + b²) in two ways (3³ + 9² and 2³ + 10²). 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
= 0⁰ x 1¹ x 2² x 3³
= 3³ + 3³ + 3³ + 3³ = 6² + 6² + 6²
= 12 x 9 and 1 + 0 + 8 = 9
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16
= 3² + 5² + 5² + 7² (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.

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)
109² = 60² + 91² (Pythagorean triple)

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 integer-sided 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

= 5² + 6² + 7² (3 consecutive squares)
= 11² - 11¹ (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.

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 n-Queens Problem for n = 6.

= 3 x 37
= 12 x 9 + 3
= 16 + 17 + 18 + 19 + 20 + 21
= (1 + 2 + 3 + ... + 34 + 35 + 36) / 6
= 20² - 17² = 56² - 55² (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 slow-moving animal. This paradox is based on a false assumption: it does not take into account that an infinite progression can have a finite sum.

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 integer-sided 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)
= 2⁴ x 7
= 11² - 3² (difference between two squares)
= 2⁷ - 2⁴ (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.

is a permutable prime.
355/113 = 3.1415929... A good approximation of .

There are 13 consecutive divisible integers (non-primes) between 113 and 127.

A113 is an inside joke present as an Easter egg in animated films created by alumni of California Institute of the Arts, referring to the classroom number used by graphic design and character animation students at the school that was used by John Lasseter and Brad Bird among others. Brad Bird first used it for a license plate number in the "Family Dog" episode of Amazing Stories. It has since appeared in other Disney movies and every Pixar movie. -- Source Wikipedia

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 Saint-Cergue, 1260 Nyon, Switzerland.

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.

is a heptagonal pyramidal number and a lucky number.
is the number of 8-vertex rooted trees.

= 23 x (2 + 3)
= 5! - 5

is the lowest value of the longest edge in a Heronian Tetrahedron.
117² +/- 2 are primes.

= 11³ - 2³ = 5³ - 2³

is the smallest 3-digit number having two prime digit permutations: 191 and 911.

= 17 + 19 + 23 + 29 + 31 (sum of consecutive primes)

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)

= 2² + 4² + 10²
= 3¹ + 3² + 3³ + 3⁴
= 31² - 29² = 17² - 13² = 11² – 1²
= 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.

is the only square known of the form 1+p+p³+p³+p⁴, where p is prime (121 = 3⁰ + 3¹ + 3² + 3³ + 3⁴).

121 = 38 + 83 (palindromic sum)

= 5! + 1
= 37 + 41 + 43 (sum of 3 consecutive primes)
= 11²
= 1 + 3 + 5 + 7 + 9 + 11 + ... + 19 + 21 (sum of consecutive odd numbers)

1/120 = 1/3 - 1/5 - 1/8

= 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

is a 42-gonal number.
is the 10^th Lucas number.

= ((((4)/.4)^4!)))) - 4
= 18 + 19 + 20 + 21 + 22 + 23 (sum of consecutive numbers)

123 x 2¹²³ – 1 = 1.30796035 x 10³⁹ =
= 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.

is an untouchable number.
is the smallest number such that its first 3 multiples contain the digit 2.

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 = 5¹⁺²
= 5³ = 11² + 2²

is a film size used in easy-to-use cameras.
= 126 = 6 x 21
= 2 x 3² x 7
= ₉C₄

= 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers)
= 2⁷ - 2¹ (difference of same powers)
= 4² + 5² + 6² + 7² (sum of consecutive squares)

is the 4th Mersenne prime and an autothecal number.
= 127 = 2⁷ - 1 = - 1 + 2⁷
(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)

= 1111111₂
is in fact the sum of seven powers of 2:
= 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶
= 2⁷ - 2⁰

= 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 (sum of consecutive primes)

2¹²⁷ – 1 is the 12th Mersenne prime number.

You can arrange the number 127 as a pattern of concentric hexagons.

is a happy number.

is the smallest number that can be expressed as the sum of 3 squares in 4 distinct ways:
= 1² + 8² + 8²
= 2² + 2² + 11²
= 2² + 5² + 10²
= 4² + 7² + 8²

= 1⁷ + 2⁷ = 43 x 3 (if S = 1ⁿ + 2ⁿ and S is odd, then it is divisible by 3)

= 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 (sum of the first 10 consecutive prime numbers)

Napoleon & Hitler:
They Were born 129 years apart;
They came into power 129 years apart;
They declared war on Russia 129 years apart;
They were defeated 129 years apart!

is a noncototient.
is the number of functions from 6 unlabeled points to themselves.

= 1² + 2² + 5² + 10² (sum of the squares of the first 4 divisors)

is the 6th Catalan number.
is the smallest number which is the sum of all of the 2-digit numbers that can be formed with its digits:
132 = 13 + 32 + 21 + 31 + 23 + 12

= 13 + 17 + 19 + 23 + 29 + 31 (sum of consecutive primes)

= 12² - 12¹ (difference of same powers)

is an octogonal number: 3 x 7² – 2 x 7 (a polygonal number of the form 3n² – 2n)
is the smallest integer n for which the sum of the proper divisors of n divides phi(n).

= 11⁰ + 11¹ + 11² (sum of powers of 11)

_{= 2}2²_{+ (2 x 3)}2 _{+ 3}2² _{- G. Sarcone}

= 2³ + 5³ (sum of cubes)

It takes 133 years before the same phases of the moon occur on the same day of the year and also the same day of the week.

134² - 67² = 13467
= 1³ + 2³ + 5³ (sum of cubes)
= ₈C₁ + ₈C₃ + ₈C₄

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 fine-structure 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).

= 4² + 11²
= 2⁷ + 2³ + 2⁰
= 10001 / 73
137² = 88² + 105² (Pythagorean triple)

137² = 18769, but 133² = 17689, 281² =78961 and 286² = 81796, four different squares having the same digits in the solution!

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!

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

= 1² + 2² + 3² + 4² + 5² + 6² + 7²

140 liters of water are needed to produce 1 cup of coffee (125 ml)!

is a Cullen number (n x 2ⁿ + 1 is a Cullen prime when n = 141).

= 2³ + 2³ + 5³ (sum of cubes)

is the smallest quasi-Carmichael number in base 8.

= 11 x 13 (product of twin primes)
= 1001 / 7
= 2² + 3² + 3² + 11²
= - 3⁴ - 4⁴ - 5⁴ - 6⁴ + 7⁴
= 43 + 47 + 53 = 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum of consecutive primes)

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:
144⁵ = 27⁵ + 84⁵ + 110⁵ + 135⁵

Reversing 12 and 144 in the equality 12² = 144 gives 441 = 21².

144 = (1 + 4)! + 4! (autothecal number)

= 71 + 73 (sum of consecutive primes)
= 3! x 4! = 4! + 5!
= 4² + 2 x 8²
= 6² + 3 x 6²

One of the largest numbers, that is not a power of 10, that has a specific name: 'gross' (= dozen dozen).

is a centered square number and also a pentagonal number.
is a product of 2 distinct primes: 5 x 29

is a factorion:
= 1! + 4! + 5! (the largest known factorion is 40'585)

= 4³ + 3⁴
= 12² + 1² = 8² + 9² (sum of squares)
= (1² + 2²)(2² + 5²)
= 3³ + 3³ + 3³ + 4³ (sum of cubes)

is an octahedral number. An octahedral number is of the form (2n³ + n)/3.

= 222 in base 8
= 5² + 11²
= (1² + 1²)(3² + 8²)

146² + 1 is prime

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

is an heptagonal number.
is the number of 6-vertex perfect graphs.

is the concatenation of the first 3 positive squares.

= 6² + 7² + 8² (sum of consecutive squares)
= 2³ + 2³ + 2³ + 5³ (sum of cubes)
= 23 + 43 + 83 (sum of 3 primes which are not sum of two squares)

150

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].

2¹⁵⁰ – 3 and 2¹⁵⁰ – 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.

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.

= 2² + 7² + 7² + 7² (sum of prime squares)

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.
= 3² x 17
= 1³ + 5³ + 3³ (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!
= 3² + 12²
= 77² - 76²
153 = 3 x 51
153 / 999 = 0.153153153... (cyclic number)

The sum of digits of 153 is a perfect square:
1 + 5 + 3 = 9 = 3²
The sum of aliquot divisors of 153 is also a perfect square:
1 + 3 + 9 + 17 + 51 = 81 = 9²

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

It is an Armstrong number, that is an integer such that the sum of the cubes of its digits is equal to the number itself. Start with any three-digit 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. OK, let's try with the number 162:
1³ + 6³ + 2³ = 225;
2³ + 2³ + 5³ = 141;
1³ + 4³ + 1³ =66;
6³ + 6³ = 432;
4³ + 3³ + 2³ = 99;
9³ + 9³ = 1458;
1³ + 4³ + 5³ + 8³ = 702;
7³ + 2³ = 351;
et voilà
3³ + 5³ + 1³ = 153
-- posted by Patrick Vennebush

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 504² = 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).

is a nonagonal number and the smallest integer which is palindromic in bases 6, 8, and 9.

is the sum of the primes between its smallest and largest prime factor.

is the number of 6-vertex 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

is a balanced prime number, that is, a prime number that is equal to the arithmetic mean of the nearest primes above and below:
p_n = (p_n-1 + p_n+1)/2

The first few balanced prime numbers are: 5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 (see the complete list in OEIS)

is the smallest 3-digit number whose square contains the same digits as its successor:
157² = 24649 and 158² = 24964
Numbers having the same peculiarity are: 13, 157, 913, 4513, 14647, 19201, 19291, 19813, 20191, 27778, ... (see the complete list in OEIS)

CIVIL is the anagram of the Roman numeral CLVII (157).

is the number of planar partitions of 11.

= 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum of consecutive odd numbers)

is the number of isomers of C₁₁H₂₄.

= 47 + 53 + 59 (sum of consecutive primes)
= 2² + 3² + 5² + 11² (sum of squares)

is the number of 9-iamonds.

is a lucky prime and the largest Heegner Number.

≈ 69/(π - e)
≈ 2⁵·log_e163

Any computation involving 163 gives an answer that is close to an integer:

= 2³ + 3³ + 4³ + 4³ (sum of cubes)
= 1 + 2 x 3⁴

= 2² x 41

= ₁₁C₃ = ₁₁C₈

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 upside-down (991), and reversed-then-turned-upside-down (199).

166!-1 is a factorial minus one prime.

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.

n⁷ - 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".

is a square whose digits are non-decreasing.

169 = 13² and in reverse order 31² = 961
= 13 + 17 + 19 + 23 + 29 + 31 + 37 (sum of consecutive primes)
= 5² + 12²
= 8³ – 7³ (difference of consecutive cubes)
= ( 3² + 2²)(3² + 2²)
= (16 + 9) + (16 x 9)

169² = 119² + 120² (Pythagorean triple)

169 = 19 - 6

is a palindromic and triangular number.

= 2³ + 2³ + 3³ + 4³ + 4³ (sum of cubes)

is a prime number.
is the sum of 3 consecutive primes: 53 + 59 + 61

= 2² + 13²

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.

is the smallest number n>1 such that n⁶+6 is prime.
= 1¹ + 7² + 5³

is happy number and an octagonal and pentagonal number.
is the number of possible partitions of the number 15.

= 2³ + 2³ + 2³ + 3³ + 5³ (sum of cubes)

is a 60-gonal number.
is the number of 7-edge graphs.
is the 'magic constant' of the Chen prime magic square:

= 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 (sum of the first 15 consecutive primes)

is a 31-gonal number.

Strangely enough, 178 and 196 are related... In fact, 178 has a square with the same digits as 196:
178² = 31,684
196² = 38,416
178 has also a cube with the same digits as 196:
178³ = 5,639,752
196³ = 7,529,536

= 2² + 2² + 7² + 11² (sum of consecutive square primes)

is a prime number.

3² + 691² + 2293² = 179³

is the total number of degrees in a triangle.

arctan1 + arctan2 + arctan3 = 180

is a strobogrammatic prime. It can be read the same upside down or when viewed in a mirror.

181² = 105³ - 104³ (solution of x² = y³ - z³)

is a palindromic and undulating number, if written in the ternary, the negaternary, or the nonary numeral systems.

= 3³ + 3³ + 4³ + 43 (sum of cubes)
= 14² - 14

is a central polygonal number, n² - 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 = 428²

is 111 in base 13.

= 32² - 29² = 92² - 91² (difference of squares)
= 2² + 3² + 7² + 11² (sum of 4 distinct square primes)

λοπαδοτεμαχοσελαχογαλεοκρανιολειψανοδριμυποτριμματο-
σιλφιοκαραβομελιτοκατακεχυμενοκιχλεπικοσσυφοφαττο-
περιστεραλεκτρυονοπτοκεφαλλιοκιγκλοπελειολαγῳοσιραιο-
βαφητραγανοπτερύγων transliterated as: Lopadotemachoselachogaleokranioleipsanodrimhypo
trimmatosilphioparaomelitokatakechymenokichlepi
kossyphophattoperisteralektryonoptekephalliokigklo
peleiolagoiosiraiobaphetraganopterygon is a 183-letter 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 non-leap year (because there are exactly 182 days before and 182 days after).

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 22-digit 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)

= 2³ x 23 (concatenation of the first 2 primes)
= 41 + 43 + 47 + 53 (sum of consecutive primes)
= 25² - 21² = 47² - 45² (difference of squares)

is a 14-gonal number and a 63-gonal number.

= 89 + 97 (sum of consecutive primes)
= 2² + 2² + 3² + 13² (sum of squares)

is the lowest quasi-Carmichael 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)
= 2³ + 3³ + 3³ + 5³ (sum of cubes)

The 189th century would be the first to contain as few as five prime years: 18803, 18839, 18859, 18869 and 18899...

is the number of semigroups of order 4.

= 2² + 3² + 7² + 11² (sum of squares)

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 55² - 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

= 4³ + 5³ = (- 3)³ + 6³ (sum of cubes)

= 24 + 25 + 26 + 27 + 28 + 29 + 30 (sum of consecutive numbers)

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

= 4² + 5² + 6² + 7² + 8² (sum of consecutive squares)

= 1 + 2 + 3 + ... + 17 + 18 + 19 (sum of consecutive numbers)

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).

is the smallest number with 14 divisors.

= 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 (sum of consecutive primes)
= 2⁸ - 2⁶ = 4⁴ - 4³

is a twin prime with 191, and the only known odd prime p for which 2 is not a primitive root of 4p²+1.

A scalene triangle having sides 193, 194 and 195 (consecutive integers) has an area that is also an integer (16296 units²). Such triangles whose sidelengths and area are all integers are called Heronian triangles.

= (9³ - 7³) / 2

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

is a square-full number. A number is said to be square-full if for every prime, p, that divides it, p² also divides it: 196 = 2² x 7²

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 reverse-and-add 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 = 14²
169 = 13²
961 = 31²

-196 = (7 - i7)⁴ = (7 + i7)⁴ (power of complex numbers)

is the only 3-digit Keith number.

= 2³ + 4³ + 5³ (sum of cubes)
= 99² - 98²

is a 67-gonal 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=n-1 and c=n-2. 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).

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:

= 2² + 5² + 7² + 11² (sum of consecutive square primes)
= 3² + 4² + 5² + 6² + 7² + 8² (sum of consecutive squares)

= 61 + 67 + 71 (sum of 3 consecutive primes)
= 31 + 37 + 41 + 43 + 47 (sum of 5 consecutive primes)

appears in the Padovan sequence.

= 66³ – 536²

= 2³ x 5³

= 7⁰ + 7¹ + 7² + 7³

= 2³ + 4³ + 4³ + 4³ (sum of cubes)

11113² – 200² = 11313 x 10913 = 123,458,769
and the reverse of the digits of all of the above numbers (200 excepted) have the same property:
31111² – 200² = 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.

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