Zoo of Numbers: Numbers 70 to 200

What's Special About This Number?

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

is 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

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.

= 5 + 7 + 11 + 13 + 17 + 19
= 13 + 17 + 19 + 23 (sums of consecutive primes)

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

72⁵ = 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵

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

The isosceles triangle having a pair of 72° angles is called a Golden Triangle.

72 hours = 3 days

Incredible geometric sculpture made with 72 pencils by the artist and mathematician George W. Hart. [click here to enlarge]

is 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 a divisor of 10001.

73² = 48² + 55² (Pythagorean triple).

365 days in a year : 5 = 73

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

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)!

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 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
= 4x4 + 5x5 + 6x6
= 39² - 38²
= 9² - 2²

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.

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

78³ = 39³ + 52³ + 65³

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 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

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.

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.
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)

= 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!

1/81 = 0.123456790123456790123456790...
(The result is a repeating decimal containing every digit from 1-9 except for 8) - Posted by Ting Wang
1/81 seems to be the closest fraction possible to a repeating set of all the digits 0-9 placed successively. The exact value 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 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.

= 41 + 43 (sum of twin primes)
= 4¹ + 4² + 4³
= 3² + 5² + 5² + 5² (sum of cubes)

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.

is the largest known n for which 1²+2²+3²+...+n² = 1+2+3+...+m has a solution.
85/38 is a good approximation of 5.

= 2² + 9²
= 6² + 7² (sum of 2 consecutive squares)
= 4⁰ + 4¹ + 4² + 4³ (sum of powers of 4)
85² = 13² + 84² = 36² + 77²

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 the only number known whose square has no isolated digits.
A piano has 88 keys.
= 23² - 21²
= 13² - 9²
It can be read the same upside down or when viewed in a mirror.

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 the number of degrees in a right angle.

= 1 x 2 x 3² x 5
= 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 (sum of the first 9 consecutive even numbers)
= 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive primes)
= 2² + 3² + 4² + 5² + 6² (sum of consecutive squares)
= 9¹ + 9² = 10² - 10¹

Sturgeon's Law: "90% of everything is rubbish"

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.

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)

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.

= 1² + 2² + 3² + 9² (sum of squares)
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15
= 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.

= 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 + 89
= 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)

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° Celsius is the boiling point of water.

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

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

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.

= 2² + 5² + 5² + 7²

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

= 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 the largest number n known such that n - 2^k is prime for k>1. - Paul Erdös

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 number that is the product of two different substrings.

= 5² + 6² + 7² (3 consecutive squares)
= 2 x 55 = 5 x 22
= 1x2 + 2x2 + 3x2 + ... + 8x2 + 9x2 + 10x2

is also known as "eleventy" according to the number naming system invented by J. R. R. Tolkien.

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

is an abundant number and a heptagonal number.
is the side of the smallest square that can be tiled with distinct integer-sided squares.

= 2⁴ x 7

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

There are 13 consecutive divisible integers (non-primes) 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 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 number.
is the smallest number to appear 6 times in Pascal's triangle.
= 5!

120 liters of water are needed to produce 1 glass of wine (125 ml)!

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)

= 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 the 10^th Lucas number.

= 18 + 19 + 20 + 21 + 22 + 23

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.

is a Mersenne prime number.
= 127 = 2⁷ - 1

is the sum of seven powers of 2:
= 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶
You can arrange the number 127 as a pattern of concentric hexagons.

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

= 2³ + 5³

divides 11,111,111.
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).

= 10001 / 73

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.

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

144 = (1 + 4)! + 4!

144⁵ = 27⁵ + 84⁵ + 110⁵ + 135⁵

= 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

= 1! + 4! + 5!
= 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)

is a triangular number:
= 1 + 2 + 3 + 4 + 5 + ... + 15 + 16 + 17
= 3² x 17
= 1³ + 5³ + 3³
= 1! + 2! + 3! + 4! + 5!
= 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

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

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 a prime number and the largest known number whose square contains the same digits as its successor:
157² = 24649 and 158² = 24964

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

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

= 2² x 41

= ₁₁C₃ = ₁₁C₈

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 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 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 the smallest integer n so that n concatenated with n+1 is square.

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.

is a Kaprekar constant in base 3.

= 2³ x 23

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 a Kaprekar constant in base 2.

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

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 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 7,326... If we reverse it's digits and add it to the original number, we get 13,563. Like so:
7326 + 6237 = 13563
Since this new number is not a palindrome, we can continue the reverse-and-add process and it may become one:
7326 + 6237 = 13563
13563 + 36531 = 50094
50094 + 49005 = 99099 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.

= 2² x 7²

is the only 3-digit Keith number.

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

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)

is a prime number.
Adding 210 to 199 over and over again, you get 8 more primes that can be arranged together into a 3x3 magic square:

is an 'invertible prime' (turned upside down, a different prime is formed: 661).
is also a 'permutable prime', meaning that 919 and 991 are primes as well.
is the 11^th Lucas number.

= 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)!

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