Zoo of Numbers: Numbers 13 to 23

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What's Special About This Number?

Fun facts on numbers (and their peculiar 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 Devil's dozen, or a baker's dozen, also known as a long dozen (one more than a proper dozen).
is the only integer that is, along with its fourth power, the sum of two consecutive squares:
13 = 2² + 3² and 13⁴ = 119² + 120²

appears for the first time at 111th place in decimal expansion of (Pi):
3.141592653589793238462643383279502884197169399375105820974944
592307816406286208998628034825342117067982148086513...

is the number of Archimedian solids:

is the sum and the difference of 2 consecutive squares:
13 = 2² + 3² = 7² - 6²

13 = (-1³+ 3³)/2

13² = 5² + 12² = 85² - 84² (Pythagorean triple)
13² = 8³ - 7³ (Diophantine equation)
13² = 7 + 8 + 9 + ... + 17 + 18 + 19 (sum of consecutive numbers)

13² divides 12! + 1

1 + 2 + 3 + 4 + ... + 10 + 11 + 12 + 13 = 1² + 2² + 3² + 4² + 5² + 6²

If we subtract the sum of its digits, we get a perfect square:
13 - (1 + 3) = 9 = 3²
If we add the product of its digits, we get another perfect square:
13 + (1 x 3) = 16 = 4²

(10⁶ - 1)/13 = 27 x 37 x 77 and 77 - 37 - 27 = 13
- by Gianni A. Sarcone -

1/13 = 0.076923076923076923... (rational periodic infinite decimal)
On multiplying the periodic number 76923 by successive multiples of 13, this curious numerical pattern is obtained:

A scalene triangle having sides 13, 14 and 15 (consecutive integers) has an area that is also an integer (84 units²). Such triangles whose sidelengths and area are all integers are called Heronian triangles.
- For more info on the Heronian triangles read the interesting article by M. P. Cohen "Generating Heronian Triangles with consecutive Integer Sides" in Journal of Recreational Mathematics, Vol. 30(2) 121-124.

is the smallest prime that can grow 6 times by the right:
13 is prime,
139 is prime,
1399 is prime,
13999 is prime,
139991 is prime,
1399913 is prime,
13999133 is prime.

13,333,333,333,333 / 13 = 1,025,641,025,641 (prime number!)

Palindromic equalities involving 13:
13 x 62 = 26 x 31
13 x 93 = 39 x 31
13 x 13 = 169 and 961 = 31 x 31

(13 - 1)! + 1 = 0 (mod 13²)

There are two ways to pack neatly thirteen (13) circular or spherical items in a rectangular space:

The numerical pattern abcabc is divisible by 13.

p¹² - q¹² is divisible by 13 only when p and q are not divisible by 13.

The sum S(n) = 1⁴ⁿ⁺² + 2⁴ⁿ⁺² + 3⁴ⁿ⁺² + 4⁴ⁿ⁺² + 5⁴ⁿ⁺² + 6⁴ⁿ⁺² is divisible by 13 for any positive integer.

A torus (kind of donut) can be sliced into 13 pieces with just 3 plane cuts.

Number anagram: ELEVEN + TWO = TWELVE + ONE

While the Earth revolves once, the Moon revolves 13 times.

The US flag displays 13 stripes - for the original 13 states.

The average human bladder can hold 13 ounces of liquid.

'Magicicada' life cycles
Periodical cicadas spend five juvenile stages in their underground burrows, and during their 13 or 17 years underground they grow from approximately the size of a small ant to nearly the size of an adult. In the spring of their 13th or 17th year, a few weeks before emerging, the nymphs construct exit tunnels to the surface. These exits are visible as approximately 1/2 inch diameter holes, or as chimney-like mud "turrets" the nymphs construct over their holes. Sometimes a large proportion of the cicada population emerges in one night!

Probability
There is always at least one Friday 13^th in each year.
The probability of being born on a Friday the 13th is about 1/214... OK, but which day of the week (Sunday, Monday, etc.) is the probability largest to fall on the 13th of a random month, in a random year? (Or is this probability the same for each day of the week?)
No, the probability that the 13th of a certain month in a certain year is a Friday, is slightly higher (at 688/4800). The probability that the 13th of a certain month in a certain year is a Thursday or a Saturday, is the lowest (at 684/4800).
- posted by Patrick Freymond -

Any month that starts on a Sunday will have a Friday the 13th.
Any non leap year February that starts with a Sunday, there will be a Friday the 13th in February and March...

According to historians, there were 13 people at Christ's Last Supper and Christ was crucified on Friday 13th. So, Friday 13th is considered as unlucky... But thirteen may be considered a 'bad' number simply because it is one more than 12, which is a popularly used number in many cultures (due to it being a highly composite number). When a group of 13 objects is divided into two, three, four or six equal groups, there is always one object leftover.

The fear of the number 13 is called 'Triskaidekaphobia'. A specific fear of Friday the 13th is called Paraskavedekatriaphobia (from Greek Παρασκευή Friday + δεκατρείς thirteen + phobia) or Friggatriskaidekaphobia. In the Spanish-speaking world, it is Tuesday the 13th that brings bad luck... A proverb runs "En martes, ni te cases ni te embarques" (on Tuesday, neither get married nor start a journey).

"13bis, est-ce un nombre pair ou impair?" (13bis - French road numbering scheme - is an even or an odd number?) -- R.Queneau, 'Le Vol d'Icare'

"Tirè el trédas" (to draw the thirteen) means to die in Piedmontese.

THIRTEEN in different languages (© G. Sarcone) (Old English thrêotêne)

Indo-european | *TRÍDEKM(T) | Sanskrit | trayodaśa | Greek, Attic | τρεισκαίδεκα TRESKAIDEKA | Latin | TREDECIM |

Italian tredici; French treize; Spanish trece; Portuguese treze; Catalan and Provençal tretze; Romanian treisprezece (treişpe, informal); Rumansh tredisch; Sardinian tréchi.

Old Germanic | THRI(S)TEHUN |

Dutch dertien; German dreizehn; Danish and Norwegian tretten; Swedish tretton; Icelandic þrettán.

Old Slavic | TRINADESETJE |

Russian тринадцать trinadtsat'; Czech třináct; Slovenian trinajst; Polish trzynaście.

Chinese 十三 (complex form 拾参) shísān.

Ancient Egyptian met'khemte; Akkadian shalash-sher(et).

Arabic ثلاثة عشر thalatha'ashar; Hebrew שלוש עשרה shlosh'esrei; Maltese: tlettax; Amharic asra-sost.

More languages Magyar tizenhárom. Mayan oxlahun. Nahuatl mahtlāctli onēyi. Suomi kolmetoista. Zulu ishumi na tatu.

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is a product of 2 primes (2x7) and a repfigit number.
is a Catalan number. Catalan numbers are generated by the formula C_n = [(2n)!/n!(n+1)!]
is a square pyramidal number:
= 1 + 4 + 9 = 1² + 2² + 3²
is the integer solution to: x² - y² - x - y = 30 [x = 16, and y = 14]

10¹⁴ - (2 x 14 - 1) and 10¹⁴ - (2 x 14 + 1) are 14-digit twin primes.

There are only 14 prime knots < order 7:

The cuboctahedron, the truncated cube, and the truncated octahedron each have 14 faces.

= 13² + 3³
= 2 + 3 + 4 + 5 (summ of consecutive integers)
= 2¹ + 2² + 2³ (summ of consecutive powers of 2)
= 7(16² - 14²)/(16 + 14)
= (log₂128)(log₇49)

1/14 = 0.0714285714285714285... Curiously 7, 14, and 28 are factors and multiples of 14. The digit 5 tells you how many digits 71428 has before they repeat!

Arithmetic curiosity: 14² – 7² = (14 + 7) x 7

The squares of 13, 14 and 31 share the same digits:
13² = 169; 14² = 196; 31² = 961

A fortnight has 14 days (i. e. two weeks, from Old English: feowertiene niht, fourteen nights).

Most people fart 14 times a day on average... ;-)

February 14 is St Valentine's day.

CALENDAR REFORM IN ENGLAND
In September 1752, Great Britain switched from the Julian Calendar to the Gregorian Calendar. In order to achieve the change, 11 days were 'omitted' from the calendar: the day after 2 September 1752 was 14 September 1752.

It takes 14 days (one fortnight) for the Moon to wax from new to full or to wane from full to new.

In ancient Egypt, Osiris - the god of the afterlife - was cut into 14 parts. Each part represented one of the 14 full moons (each year has in fact 12 to 14 full moons).

There is a Finnish word that contains a continuous sequence of 14 dots: Pääjääjää ('the main stayer' in Finnish, partitive case).

The French expression: "chercher midi à quatorze heures" (to look for noon at 14:00 - i.e. 2 pm) is a quirky way of telling someone that he is making an issue more difficult than it needs to be - that is, turning something simple into something complicated.

The 'Fourteen Words' is a phrase used by white nationalists. It refers to the 14-word slogan: "we must secure the existence of our people and a future for white children". Often found in combination with 88 (14/88, 8814...).

FOURTEEN in different languages (© G. Sarcone) (Old English fêowertêne)

Indo-european | *KWETWORDEKM(T) | Sanskrit | catúrdaśa | Greek, Attic | τέτταρες καὶ δέκα TETTARES KAI DEKA | Latin | QVATTVORDECIM |, Archaic Latin | *QVATBORDECEM |

Italian quattordici; French, Portuguese quatorze; Spanish catorce; Catalan and Provençal catorze; Romanian paisprezece (< patru-spre-zece); Rumansh quitordisch; Sardinian batódighi.

Old Germanic | FITHWORTEHUN |

Dutch veertien; German vierzehn; Danish and Norwegian fjorten; Swedish fjorton; Icelandic fjórtján.

Old Slavic | CETYR(I)NADESETJE |

Russian четырнадцать chetyrnadtsat'; Czech čtrnáct; Slovenian štirinajst; Polish czternaście.

Chinese 十四 shísi.

Ancient Egyptian met'efte; Akkadian erbesher(et).

Arabic أربعة عشر arba'a'ashar; Hebrew ארבע עשרה arba'esrei; Maltese: erbatax; Amharic asra-arat.

More languages Magyar tizennégy. Mayan canlahun. Nahuatl mahtlāctlinnāhui. Suomi neljätoista. Zulu ishumi na ne.

All prime quadruplets - with only two exceptions - enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, e.g.: 11, 13, 17, 19.

is the number of 3-digit palindromic primes: 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, and 929.

is a triangular number: 1 + 2 + 3 + 4 + 5
is the positive solution to: 2x² - 27x = 45

15 and 21 are the smallest pair of triangular numbers whose difference and sum are also triangular: 21 - 15 = 6, and 21 + 15 = 36

= 3log₂32

= 4! - !4 (substraction of factorial and subfactorial)

2⁴ⁿ – 1 is divisible by 15.
3n⁵ + 5n³ + 7n is divisible by 15.

= 8² - 7²
= 4² - 1²

15² = (1 + 2)(3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12)
15⁴ = 4⁴ + 6⁴ + 8⁴ + 9⁴ + 14⁴

15,15 ≈ e^e

The simpliest magic square having a 'magic constant' of 15:

is the name of a famous slide puzzle: the 15 puzzle.

is the number of 'The Devil' in the major arcana of Tarot cards.

A penguin swims at a speed of approximately 15 miles per hour.

In 2007, Danish sperm donors have helped to engender an estimated 15,000 children around the world!

15.23 is the hour of temptation. In fact, experts say it is the time when it is most likely person to encroach on sweets... More info at dietplanfortoday.com.

The only 15-letter word that can be spelled without repeating a letter is "uncopyrightable".

In the Hebrew numeral system, the numbers 15 and 16 are represented as (9+6) and (9+7) respectively, instead of (10+5) and (10+6). This is done in order to refrain from using the sacred combinations, called tretragrammaton, that are a part of the name of God in Judaism.

Huli, a Papuan language spoken by the Huli people of the Southern Highlands province of Papua New Guinea, is known to have a quindecimal or pentadecimal (base-15) numeral system. For instance, the Huly people say "93" with ngui waraga, ngui kane-gonaga tebira, which means "(15 x 6) + (3 objects of the seventh 15)". The number system of Huli.

FIFTEEN in different languages (© G. Sarcone) (Old English fîftêne)

Indo-european | *PENKWEDEKM(T) | Sanskrit | pañcadaśa | Greek, Attic | πεντεκαίδεκα PENTEKAIDEKA | Latin | QVINDECIM |

Italian quindici; French, Portuguese, Catalan, and Provençal quinze; Spanish quince; Romanian cincisprezece (cinşpe, informal); Rumansh quendisch; Sardinian bíndhichi.

Old Germanic | FIMFITEHUN |

Dutch vijftien; German fünfzehn; Danish and Norwegian femten; Swedish femton; Icelandic fimmtán.

Old Slavic | PEnTINADESETJE |

Russian пятнадцать pyatnadtsat’ ; Czech patnáct; Slovenian petnajst; Polish piętnaście.

Chinese 十五 (complex form 拾伍) shíwǔ.

Ancient Egyptian metti'ue; Akkadian hamish-sher(et).

Arabic خمسة عشر khamsa'ashar; Hebrew חמש עשרה khamesh'esrei; Maltese: ħmistax; Amharic asra-amist.

More languages Magyar tizenöt. Mayan holahun. Nahuatl caxtōlli. Suomi viisitoista. Zulu ishumi na nhlanu.

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is the only integer of the form x^y=y^x with x y (2⁴=4²).
is the smallest number with exactly five divisors.

= 1 + 3 + 5 + 7 (sum of the four first odd numbers)
= 128^4/7

16 + 9 = 25 = 5²
16 x 9 = 144 = 12²

Mathematically correct result derived by incorrect calculation method (anomalous cancellation):
16 = 16 = 1
64    64   4

The base 16 notational system for representing real numbers, called hexadecimal, is particularly important and used extensively in computer science, since four bits (each consisting of a 'one' or 'zero') can be succinctly expressed with just a single hexadecimal digit. The 16 distinct digits used to represent numbers in hexadecimal notation are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

The human eye sees continuous motion at between 16 and 20 frames per second.

There are 16 orders of mammals.

Romanian constructs the names of the numbers 11 to 19 by a regular pattern which could be translated as 'one-over-ten', 'two-over-ten', etc. All the other Romance languages use a pattern like 'one-ten', 'two-ten', etc. for 11 to 15; and the pattern 'ten-and-seven', 'ten-and-eight', 'ten-and-nine' for 17 to 19. For 16, however, they split into two groups: some use 'six-ten', while others use 'ten-and-six':
• 'Sixteen': French seize, Italian sedici, Catalan and Occitan setze, Sardinian sédichi.
• 'Ten and six': Portuguese dezasseis or dezesseis, Spanish dieciséis.
• 'Six over ten': Romanian şaisprezece (where spre derives from Latin 'super').

SIXTEEN in different languages (© G. Sarcone) (Old English siextiêne)

Indo-european | *SEKSDEKM(T) | Sanskrit | ṣoḍaśa | Greek, Attic | ἑκκαίδεκα HEKKAIDEKA | Latin | SEDECIM |

Italian sedici; French seize; Catalan, and Provençal setze; Spanish dieciséis (< diez y seis); Portuguese dezasseis or dezesseis; Romanian şaisprezece (saişpe, informal); Rumansh sedisch; Sardinian sédichi.

Old Germanic | SEKSTEHUN |

Dutch zestien; German sechzehn; Danish and Norwegian seksten; Swedish sexton; Icelandic sextán.

Old Slavic | ShESTINADESETJE |

Russian шестнадцать shestnadtsat’ ; Czech šestnáct; Slovenian šéstnajst; Polish szesnaście.

Chinese 十六 (complex form 拾陆) shíliù.

Ancient Egyptian mettise; Akkadian shesh-sher(et).

Arabic ستة عشر sitta'ashar; Hebrew שש עשרה shesh'esrei; Maltese: sittax; Amharic asra-sidist.

More languages Magyar tizenhat. Mayan uaclahun. Nahuatl caxtōloncē. Suomi kuusitoista. Zulu ishumi ne situpa.

17 and 19 form a twin prime pair.
is the 7^th prime number and the only prime of the form p^q + q^p, where p and q are prime: 17 = 2³ + 3²
is also the only prime number which is the sum of 4 consecutive primes:
17 = 2 + 3 + 5 + 7
is the smallest number with four representations as a sum of three primes: 17 = 2 + 2 + 13 = 3 + 3 + 11 = 3 + 7 + 7 = 5 + 5 + 7
is the lowest number that can be written as A³ + B² in 2 distinct ways:
17 = 2³ + 3² and 17 = 1³ + 4²

is a 'cult number' - a number which has an unusually large number of fans. Cult numbers are related to 'psychologically random numbers', i.e. numbers which are chosen more often when someone is asked to pick a random number.

is the lowest possible number of givens for a Sudoku puzzle (数独) with a unique solution (this was proven in 2012 by Gary McGuire, Bastian Tugemann, and Gilles Civario).

abcdefghabcdefgh is divisible by 17 (replace each letter with a digit. E.g. 1234567812345678). The reason is that any number of that form is a multiple of 100,000,001 - which is divisible by 17. - David Grossberg

The multiplicative inverse of 17 gives a decimal expansion with a 16-digit repeating decimal:
1/17=0,05882352941176470588235294117647...

17 = 9² - 8² = 3⁴ - 4³

n² + n + 17 = Prime, for n = 0 to 15

17 x 65,359,477,124,183 = 1,111,111,111,111,111

17² can be expressed as the sum of 1, 2, 3, 4, 5, 6, 7, 8 distinct squares.

17² = 8² + 15² (primitive Pythagorean triple)
17² = 1³ + 2³ + 4³ + 6³
17² = 0⁰ + 1¹ + 2² + 3³ + 4⁴
17³ = 47² + 52² = 17² + 68²
17³ = 4,913 and 4 + 9 + 1 + 3 = 17
17⁴ = 1⁴ + 4⁴ (first sum of 2 distinct fourth powers)

17³ = 4,913 = (4 + 9 + 1 + 3)³
17⁴ = 83,521 = (8 + 3 + 5 + 2 – 1)⁴
17⁶ = 24,137,569 = (24 – 13 + 75 – 69)⁶

_{= 2}2²_{+ 1 (Fermat prime)}

A regular 17-gon can be constructed by using rule and compasses only.

There are 17 ways to express 17 as the sum of one or more primes; 17 is the only integer that is equal to the number of prime partitions of itself.

There are almost 17 ounces in a pound.

The Alhambra, a Moorish castle in Spain contains 17 different tiling patterns, which is actually the total number of possible tilings by translating and rotating geometric patterns. These 17 possible plane symmetry groups are better known as wallpaper groups (see diagram below).

There are seventeen 17 in the first thousand digits of pi's decimal expansion.

A hypercube must be cut along 17 faces to unfold into a three-dimensional cross.

Magic triangles: the digits 1 through 9 are arranged on a triangle so that there is one digit on each vertex and two digits on each side between the vertices. The total of the four digits on any given side must equal 17.

Plutarch (Greek: Πλούταρχος), the most famous biographer of the ancient world, records "The Pythagoreans also have a horror for the number 17, for 17 lies exactly halfway between 16, which is a square, and the number 18, which is the double of a square, these two, 16 and 18, being the only two numbers representing areas for which the perimeter equals the area (16 = 4 + 4 + 4 + 4 = 4 x 4 and 18 = 3 + 3 + 6 + 6 = 3 x 6)".

The famous problem of the 17 camels: a sheik has 3 children and owns 17 camels. His will stipulates that the eldest is to receive half his property; the second son is to receive the third of his property; and the third one, the ninth of his property. On his death, how would the sons share out the inheritance? Solution: They borrow a camel, share out, and give back the camel: (17+1)/2 + (17+1)/3 + (17+1)/9 = 17 camels
(ref. Gaston Boucheny, "Curiosités et Récréations Mathématiques". Paris, 1939)

Another well-known problem involving 17: x and y are integers between 2 and 100. Alex knows S = x+y, and Pat knows P = xy, but they do not know x and y. "I can't calculate them" Pat says, "I knew" Alex says. "So I know these two numbers" Pat says, "In this case, so do I" Alex concludes. Solution: S = 17 = 13 + 4

Autoreferential growth rate: an enterprise growing at the rate of exactly 17.65717% per year will become exactly 17.65717 times larger in exactly 17 years, seven months, 29 days, and 21 hours... Which is exactly 17.65717 years! - Source: niquette.com.

On 'Yellow Pigs Day' (July 17) in 1717, Handel's Water Music was first performed.

is the average brain weight of a marmot (in grams).

Number 17 is a 1932 crime film directed by Alfred Hitchcock. It was based on the play 'Number 17' by Joseph Jefferson Farjeon. The film is about a group of criminals who committed a jewel robbery and put their money in an old house over a railway leading to the English Channel.

A day on Uranus lasts for 17 hours.

17 is the number of syllables in a haiku. A haiku consists of three line; 5 syllables in the first one, 7 syllables in the second and finally 5 syllables in the third line.

March 17th is Saint Patrick Day, the patron Saint of Ireland. A common symbol is the three-leaved shamrock. It is told that Saint Patrick used the shamrock to explain the Holy Trinity.

Most Italians believe that 17 is unlucky, and they are very superstitious concerning it! For instance, Alitalia airplanes have no 17th row [editor's note: are you sure?], Italian buildings do not have a seventeenth floor (in some hotels and apartment buildings the room number '17' is replaced with '16bis'), and when the Renault R17 went to Italy, it's name was changed to R117. Part of the reason that Italians do not like 17 comes from their Latin heritage. In fact, XVII rearranged spells VIXI, which means "I am defunct (dead)" in Latin. The fear of the number 17 is called 'heptadecaphobia' or 'heptakaidekaphobia' (see below).

17 is a swear word in Swedish. The origin is still debated, and is commonly used as "sjutton också!" (seventeen, too!).

SEVENTEEN in different languages (© G. Sarcone) (Old English seofontiêne)

Indo-european | *SEPTEMDEKM(T) | Sanskrit | saptadaśa | Greek, Attic | ἑπτακαίδεκα HEPTAKAIDEKA | Latin | SEPTEMDECIM (or SEPTENDECIM) |

Italian diciassette (< dece ac septe); French dix-sept; Spanish diecisiete (< diez y siete); Portuguese dezassete or dezessette; Catalan disset; Provençal dètz-e-sèt; Romanian şaptesprezece (şapteşpe, informal); Rumansh gissiat (siat = 7); Sardinian decassète.

Old Germanic | SIBUNTEHUN |

Dutch zeventien; German siebzehn; Danish and Norwegian sytten; Swedish sjutton; Icelandic sautján.

Old Slavic | SEDMINADESETJE |

Russian семнадцать semnadtsat’ ; Czech sedmnáct; Slovenian sédemnajst; Polish siedemnaście.

Chinese 十七 (complex form 拾柒) shíqî.

Ancient Egyptian metsafkhe; Akkadian shebe-sher(et).

Arabic تسعة عشر sab'a'ashar; Hebrew שבע עשרה shva'esrei; Maltese: sbatax; Amharic asra-sebat.

More languages Magyar tizenhét. Mayan uuclahun. Nahuatl caxtōlomōme. Suomi seitsemäntoista. Zulu ishumi ne sikhombisa.

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is the area and also the perimeter of a rectangle with 6 and 3-unit sides.
is the only number that is twice the sum of its digits.

= 3 + 4 + 5 + 6 (sum of consecutive numbers)
= 3³ - 3²
= (2 + 8)²
= CM/L

18 ₌ 9 + 9
81    9 x 9

18² = 6² + 22² - 14² = 7² + 30² - 25²

18³ = 5832 and 5 + 8 + 3 + 2 = 18

18³ =    5832
18⁴ = 104976 (both results together include all digits once)

18 equilateral triangles make a hexagon ring.

is the solution to the classic puzzle: "my age today is three times what it will be three years from now minus three times what my age was three years ago. How old am I?" [x = 3(x + 3) - 3(x - 3)].

is the number of people it takes to consume the amount of oxygen created by an acre of trees each year.

is approximately the numbers of years of one Saros cycle. The Saros cycle is an eclipse cycle with a period of 223 synodic months (near 18 years + 11 1/3 days), useful to predict eclipses of the Sun and Moon. One cycle after an eclipse, the Sun, Earth, and Moon return to the same relative geometry, and a nearly identical eclipse will occur. The coincidence of the Saros cycle was discovered by the Babylonian.

is a hate number symbol. The first letter of the alphabet is 'A'; the eighth letter of the alphabet is 'H'. Thus, 1 plus 8, or 18, equals 'AH', an abbreviation for Adolf Hitler. Neo-Nazis use 18 in tattoos. The number is also used by "Combat 18", a violent British neo-Nazi group that chose its name in honor of Adolf Hitler. The number 18 is also considered a number of the devil due to the fact that 6+6+6=18, and 666 is the Number of the Beast.

On Earth, the length of a day 900 million years ago was only about 18 hours. - Science magazine

Another interesting Finnish word that contains a sequence of 18 dots: Pääjäävääjää ('the main waiver' in Finnish, partitive case). - posted by Juhani Sirkiä -

DIECIOCHO (= 18), is the largest Spanish number written in capital letters that has horizontal symmetry (i.e. the top is a mirror image of the bottom).

The 18-letter words 'conservationalists' and 'conversationalists' are anagrams of each other. They are the longest pair of anagrams with such properties.

There are 18 chapters in the Hindu epic tale Bhagavad Gita.

A fetus develops fingerprints at 18 weeks.

According to the system of 'gematria', the word for "life" in Hebrew (see chai) is composed of two letters whose numerical value is 18. For this reason, 18 is a lucky number in Judaism, and many Jews give gifts of money in multiples of 18 for good luck.

EIGHTEEN in different languages (© G. Sarcone) (Old English e(a)htatêne)

Indo-european | *OKTODEKM(T) | Sanskrit | aṣtadaśa | Greek, Attic | ὀκτωκαίδεκα OKTÔKAIDEKA | Latin | OCTODECIM or DVODEVIGINTI |

Italian diciotto (< dece ac octo); French dix-huit; Spanish dieciocho (< diez y ocho); Portuguese dezoito; Catalan divuit; Provençal dètz-e-uèch; Romanian optsprezece (optişpe, optîşpe, or opşpe, informal); Rumansh schotg (otg = 8); Sardinian decheòto.

Old Germanic | AHTÔTEHUN |

Dutch achttien; German achtzehn; Danish and Norwegian atten; Swedish arton or aderton; Icelandic átján.

Old Slavic | OS(T)MINADESETJE |

Russian восемнадцать vosemnadtsat’ ; Czech osmnáct; Slovenian ósemnajst; Polish osiemnaście.

Chinese 十八 (complex form 拾捌) shíbā.

Ancient Egyptian metkhmune; Akkadian saman-sher(et).

Arabic ثمانية عشر thamaniya 'ashar; Hebrew שמונה עשרה shmone'esrei; Maltese: tmintax; Amharic asra-semment.

More languages Magyar tizennyolc. Mayan uaxaclahun. Nahuatl caxtōlonēyi. Suomi kahdeksantoista. Zulu ishumi ne sishiyagalombili.

is a prime and a repfigit number.
is the maximum number of 4^th powers needed to sum to any number.

19/7 is a good rational approximation of the constant e (Euler's number, or Napier's constant).

It is possible to arrange 12 coins into 19 rows of 3 coins each.

The integers 19, 109, 1009, 10009 are prime and the next is 1000000009.

The multiplicative inverse of 19 gives a decimal expansion with a 18-digit repeating decimal:
1/19=0,052631578947368421052631578947368421...
This repeating decimal has also an interesting property: in fact, to multiply the number 105,263,157,894,736,842 by two, you just have to move the last digit 2 to the beginning of the number: 210,526,315,789,473,684. Numbers having such properties are called parasitic numbers.

With respect to divisibility, nineteen has a very simple test: 19 divides 100a + b if, and only if, it divides a + 4b.

A number abc (= 100a + 10b + c) is divisible by 19 if, and only if, 10a + b + 2c is. Example: 475 --> 47 + 2x5 = 57 = 3 x 19, so 475 is divisible by 19.

Any number abcdefghiabcdefghi is divisible by 19. Example:
123456789123456789 / 19 = 6497725743339831

The sum 2²ⁿ+3²ⁿ+5²ⁿ is always divisible by 19 and the sum 2⁴ⁿ+3⁴ⁿ+5⁴ⁿ is always divisible by 19² for any integer n³.

19 = 1 x 9 + 1 + 9, and 29 = 2 x 9 + 2 + 9. This also works for 39, 49, 59, 69, 79, 89 and 99.

1 + 2 + 3 + ... + 17 + 18 + 19 = 19 x 10

= 1² + 3² + 3² (sum of squares)
= 10² - 9² (difference of consecutive squares)
= 3³ - 2³ (difference of consecutive cubes)

= 32 - 3² - 2²
= 53 - 5² - 3²
= 72 - 7² - 2²

19³ = 27² + 37² + 69²

= - (1!) + 2! - (3!) + 4!

= F + 4 (base 16)

Mathematically correct result derived by incorrect calculation method (anomalous cancellation):
19 = 19 = 1
95    95    5

The unique possible Magic Hexagon contains consecutive integers from 1 to 19 (see image below). The numbers in each row, and in all three directions, sum to the same magic constant: 38.

is the smallest prime that can grow 8 times by the right:
19 is prime,
197 is prime,
1979 is prime,
19793 is prime,
197933 is prime,
1979339 is prime,
19793393 is prime,
197933933 is prime,
1979339339 is prime.

19 is the largest prime number that is palindromic in Roman numerals (XIX).

Kelvin Ship Waves Constant: Wakes fan out at a constant angle of about 19.5°, no matter the speed of the vessel.

The Rhind Mathematical Papyrus (RMP) contained a problem to find x so that x + x(1/7) = 19. (A quantity plus one-seventh of it becomes 19. What is the quantity? Answer: 16 + 5/8)

Oktoberfest fact: How many steins of beer can a waitress carry at one time? In November 2008, Bavarian Anita Schwarz set a new world record when she carried 19 full beer steins (5 in either hand and 9 on top) totaling 45 kg.

19 is the number of vertical and horizontal lines of a Go board (or "goban"). In fact, the board is always slightly longer than it is wide. "Go" comes from Japanese igo 囲碁 which in turn comes from Chinese wéiqí 圍棋, meaning: "encircling game". Go is called baduk 바둑 in Korea, and cờ vây in Vietnam.

"Nineteen" is the term used to describe a worthless hand in the game of cribbage.

Number 19 plays an important role in the structure of the Quran (القرآن‎ al-qurʾān). The numerical value of the Arabic word واحد (wahid), which means "One", is 19. Thus, number 19 represents the 'Oneness of God' which is a very important doctrine in Islam.

"The Sun" is numbered with 19 in Tarot cards.

NINETEEN in different languages (© G. Sarcone) (Old English nigontȳne)

Indo-european | *NEWNDEKM(T) | Sanskrit | navadaśa or ekonavimśati (20 less 1) | Greek, Attic | ἐννεακαίδεκα ENNEAKAIDEKA | Latin | UNDEVIGINTI (20 less 1) |

Italian diciannove (< dece ac nove); French dix-neuf; Spanish diecinueve (< diez y nueve); Portuguese dezanove or dezenove; Catalan dinou; Provençal dètz-e-nòu; Romanian nouăsprezece (nouăşpe, informal); Rumansh scheniv (niv = 9); Sardinian decannòbe.

Old Germanic | NIWUNTEHUN |

Dutch negentien; German neunzehn; Danish and Norwegian nitten; Swedish nitton; Icelandic nítján.

Old Slavic | DEVETINADESETJE |

Russian девятнадцать devyatnadtsat’ ; Czech devatenáct; Slovenian devétnajst; Polish dziewiętnaście.

Chinese 十九 (complex form 拾玖) shíjiǔ.

Ancient Egyptian metp'site; Akkadian or  tishêret.

Arabic تسعة عشر tisha 'ashar; Hebrew תשעה עשרה tsha'esrei; Maltese: dsatax; Amharic asra-zethain.

More languages Magyar tizenkilenc. Mayan bolonlahun. Nahuatl caxtōlonnāhui. Suomi yhdeksäntoista. Zulu ishumi ne sishiyagalolunye.

is a tetrahedral number and a composite number (its proper divisors being 1, 2, 4, 5 and 10).
is also the number of rooted trees with 6 vertices.

is the base of the ancient Mayan numeral system. The vigesimal or 20-base numeral system is also present in some number names in many different languages, such as:
French, quatre-vingts, "four twenty" = 80; Danish, tres (short for tresindstyve), "three times twenty" = 60; Albanian, dyzet, "two twenties" = 40; Georgian, samotsdashvidi, "three-twenty-and-seven" = 67; Basque, hirurogeita hamabost, "three-score-and-ten-five" = 75; Ainu, tu hotnep, "two twenties" = 40; Yoruba (Nigerian language), métadinlaaadota, "three from ten from twenty three", i.e. "3 from [10 from (20 x 3)]" = 47; English, four score = 80. -- G. Sarcone

= 1 + 1 + 2 + 3 + 5 + 8 (sum of the first 6 Fibonacci numbers)
= 2 + 4 + 6 + 8 (sum of the first 4 even numbers)
= 1 + 3 + 6 + 10 (sum of the first 4 triangular numbers)

= 2² + 2^2x2
= 4(4/4 + 4)
= 6! / 6²

≈ e - (very very close to 20)

20 = 6² - 4²
20² = 29² - 21² = 7⁰ + 7¹ + 7² + 7³
20³ = 7³ + 14³ + 17³ = 11³ + 12³ + 13³ + 14³ (consecutive cubes)

In geometry, a regular icosahedron is a solid with an equilateral triangle on each of its 20 faces. It is not possible to make a regular polyhedron with more than 20 faces.

Magic triangle with a constant of 20. 126 is the sum of the squared numbers on any side.

20 is the number of pharmacist's drops in one milliliter.

The bamboo flowers at the end if its life cycle - which for bamboo is about every 20 to 120 years!

20 = 8 PM

1 meter corresponds approximately to 1/(20 x 10⁶) of the Earth Meridian. - posted by Hassan Taifour

20th April
At 20:02 on Wednesday, April 20, 2002, it was 20:02 20/02 2002, (palindromic date) in most European countries. 20:02 20/02 2002 is a palindromic date because it is the same read from left to right as it is read from right to left!

'20/20 vision' is a term used to express normal visual acuity (the clarity or sharpness of vision) measured at a distance of 20 feet. If you have 20/20 vision, you can see clearly at 20 feet what should normally be seen at that distance.

A set of 20 units may also be referred to as a score.

'Nebuchadnezzar' is a huge wine bottle holding the equivalent of 20 normal bottles of wine.

The word 'hussar' (obsolete; horseman of the Hungarian light cavalry) meant originally "twenty-worth". In fact, in the 16th century, the Hungarian King Mathias Corvinus decreed that each group of 20 households in a town was to furnish a horse soldier, properly a huszár ('twentieth', from Magyar húsz, 20)...

Read carefully the 20-word long sentence below:
"I do not know where family doctors acquired illegibly perplexing handwriting nevertheless, extraordinary pharmaceutical intellectuality counterbalancing indecipherability, transcendentalizes intercommunications incomprehensibleness".
This is a special sentence where the Nth word is N letters long!
e.g. the 4rd word is 4 letters long, the 13th word is 13 letters long, and so on.

In any population, there are always nearly 20 percent of people that are “contrary Mary”, that is persons marked by a disposition to oppose and contradict, who are always in contrast to any opinion taken by the majority or to opinions that are largely accepted.

There are 20 perfect interlingual homographs (words that share the same spelling and meaning in French, English, German, Italian, Spanish, etc...): album, diesel, embargo, gangster, hotel, jazz, jockey, karate, laser, libido, mafia, matador, motel, paranoia, radar, radio, revolver, sauna, taxi, telex, virus, yoga. - G. Sarcone

TWENTY in different languages (© G. Sarcone) (Old English twentig)

Indo-european | *WIK'MT(I) | Sanskrit | vimśati | Greek, Attic | εἴκοσι(ν) ÊKOSI(N) (< wikati, cf. Doric ϝίκατι) | Latin | VIGINTI |, Archaic Latin | *DVICINTI |

Italian venti; French vingt; Spanish veinte; Portuguese vinte; Catalan, and Provençal vint; Romanian douăzeci; Rumansh vegn; Sardinian bínti.

Old Germanic | *TWAI(GEN)TIGUS |

Dutch twintig; German zwanzig; Danish and Norwegian tyve; Swedish tjugo; Icelandic tuttugu.

Old Slavic | D(U)VADESETI |

Russian двадцать dvadtsat’ ; Czech dvacet; Slovenian dvajset; Polish dwadzieścia.

Chinese 二十 èrshi or 廿 niàn (complex form 貳拾 or 念).

Ancient Egyptian [ḏbˁ.ty] ḏubāʕatay; Akkadian eshra.

Arabic عشرون 'ishrun; Hebrew עשרים esreem; Maltese: għoxrin; Amharic haya.

More languages Magyar húsz. Mayan hunkal. Nahuatl cempōhualli. Suomi kaksikymmentä. Zulu amashumi amabili.

Buy your favorite Number (20) here.

is the 3rd star number (3 x 3 + 4 x 3).
is a Fibonacci number, and the smallest number of differently sized squares that are needed to tile a square (see below).
is the number of spots on a standard cubical die (1+2+3+4+5+6).
is the number of ways of choosing a set of 5 elements from a set of 7 elements.
is x for: y=2x-22 and 3x-4y+17=0

= 11² - 10² = 5² - 2² (differences of two squares)
= 1² + 2² + 4² (sum of three squares)
= 4⁰ + 4¹ + 4²
= 6² - 5² + 4² - 3² + 2² - 1²
= CV/V

= 2 + 2 + 17 = 3 + 5 + 13 = 3 + 7 + 11 = 5 + 5 + 11 = 7 + 7 + 7 (the smallest number with 5 representations as a sum of 3 primes)

If a square number ends with xyxyxyxyxy, then xy = 21, 29, 61, 69 or 84. For instance: 508,853,989² = 258932382121212121

2²¹ - 21 is prime.

21 is the fewest number of distinct integer-sided squares needed to tile a square. The simple perfect squared square shown below was discovered by A. J. W. Duijvestijn in 1976 - Journal of Combination Theory.

"When I Was One-and-Twenty" is a dramatical verse by Alfred Edward Housman about the futility of love, youth, experience, and the irony in living life.

'Twenty-one' is the name of a card game (also called 'blackjack').

According to the Mayan numerology December 21, 2012 (13.0.0.0.0, 4 Ajaw, 3 K’ank’in in Mayan reckoning) is the end of the world.
- You can purchase our nice doomsday T-Shirts here.
- More info about the Mayan prophecy.

The German 'Twenty-One Club' (Zwanzigeins e. V.) is dedicated to the promotion of a new way to spell numbers in order to make mental arithmetic easier. The supporters of the club want Germans to say zwanzigeins (twenty-one) instead of the current einundzwanzig (one-and-twenty).

TWENTY-ONE in different languages (© G. Sarcone) (Old English ān and twentig)

Indo-european | *WIK'MTI OINOS(KwE) | Sanskrit | ekavimśati (1 + 20) | Greek, Attic | εἷς καὶ εἴκοσι(ν) HES KAI EKOSI(N) (1 + 20) |,                 or | εἴκοσι (καὶ) εἷς EKOSI (KAI) HES (20 + 1) | Latin | UNUS (-A, -UM) ET VIGINTI (1 & 20) |,     or | VIGINTI UNUS, -A, -UM (20 + 1) |

Italian ventuno; French vingt-et-un; Spanish veintiuno; Portuguese vinte e um; Catalan vint-i-u; Provençal vint-e-un; Romanian douăzeci şi unu (douăzeşunu, informal); Rumansh ventgin; Sardinian bintúnu.

Old Germanic | *AINAZ (UNTA) TWAI(GEN)TIGUS (1 & 20) |

Dutch eenentwintig; German einundzwanzig; Danish and Norwegian enogtyve; Swedish tjugoett; Icelandic tuttugu og einn.

Old Slavic | D(U)VADESETI I JEDINU |

Russian двадцать один dvadtsat’ odin; Czech jedenadvacet, or dvacet jeden; Slovenian ênaindvajset; Polish dwadzieścia jeden.

Chinese 二十一 èrshíyī (complex form 貳拾壹).

Ancient Egyptian ḏubā'ataywie; Akkadian ishten (u) eshra (1 & 20).

Arabic واحد و عشرون wahed wa-'ishrun (1 & 20); Hebrew עשרים ואחת esreem ve'akhat (20 & 1); Maltese: wieħed u għoxrin; Amharic haya and.

More languages Magyar huszonegy. Mayan hunkal catac hun. Nahuatl cēmpōhualoncē. Suomi kaksikymmentäyksi. Zulu amashumi amabili na nye.

is a pentagonal number (22 = 1 + 4 + 7 + 10) and a centered heptagonal number.
is the number of partitions of 8. There are, in fact, precisely 22 ways to express 8 as a sum of positive integers: 8, 7+1, 6+2, 6+1+1, 5+3, 5+2+1, 5+1+1+1, 4+4, 4+3+1, 4+2+2, 4+2+1+1, 4+1+1+1+1, 3+3+2, 3+3+1+1, 3+2+2+1, 3+2+1+1+1, 3+1+1+1+1+1, 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, and 1+1+1+1+1+1+1+1

is the smallest number which can be expressed as the sum of 2 primes in 3 ways: 3 + 19, 5 + 17, and 11 + 11
is the smallest number of differently sized squares that are needed to tile a '2a x 1a' rectangle.
is the number of different ways of linking 5 hexagons together (pentahexes, see image below).

22/7 = 3.142... Honest approximation of .
22 = κβ (in Ionic Greek numeration)

= 4 + 5 + 6 + 7
= 1⁴ + 2³ + 3² + 4¹
= MC/L

22 x 22 = 4 x 11 x 11 = 484 is a palindromic equality!

"22 v'là les flics" is a French popular expression used to warn of the coming of the police.

The Olivetti "Lettera 22" is a portable mechanical typewriter designed by Marcello Nizzoli in 1949. It is in the permenant collection of the MOMA.

In numerology, 22 is considered a significant number, sometimes called the Master Builder or Spiritual Master in Form. Number-22 persons are said to find themselves feeling as if they live in 2 worlds, one which is overwhelmed by the mundane, and the other by the fantastic; they find the most fulfillment working on a higher plane to help people.

23 is the smallest odd prime which is not a twin prime.

is one of only two positive integers which CANNOT be written as the sum of 8 positive cubes (239 is the other one).
   --- Christopher Carl Heckmann (23 letters!)

is the smallest prime number with consecutive digits.
is the lowest prime whose reversal is a power: 32 = 2⁵ - Trigg
is the lowest number of integer-sided parallelepipeds you can fit in a box so that no two parallelepipeds share a common length.
occurs twice in the list of the fortunate numbers.

Palindromic equalities involving 23:
23 x 64 = 46 x 32
23 x 96 = 69 x 32

23! is 23 digits long.
= 5 + 7 + 11 (smallest prime that is a sum of 3 consecutive primes)
= (3 x 5) + (3 + 5)
= 3 x 2³ + (-1)³ (sum of 4 cubes)
= 2 x 2³ + 7 x 1³ (sum of 9 positive cubes)
= 12² - 11²
= - (2² - 3³)
= 1⁴ + 2³ + 3² + 4¹ + 5⁰

10²³ - 23 = 99,999,999,999,999,999,999,977 is the largest 23 digit prime.

23 x 2239 x 2393 = 123232321 (palindromic prime)

Repeat the digit 1, 23 times like this: 11,111,111,111,111,111,111,111 and you obtain a prime number.

23.140692632779269005... is e^π, also known as the Gelfond's constant.

The 'Birthday Paradox' states that a group of 23 randomly-selected persons is the lowest number where there will be a probability higher than 50 per cent that two people will share the same birthday. For 60 or more people, the probability is greater than 99 per cent!

In telegraphers code 23 means "break the line".

Each parent contributes 23 chromosomes to the start of human life. The nuclei of cells in human bodies have 46 chromosomes made out of 23 pairs.

Twenty-three is the Sacred Number (along with 5) of Eris, goddess of discord, according to the Principia Discordia; it is the number of the 'Illuminati'.

Twenty-three is one of the most commonly cited prime numbers... Numerous people have claimed to see 23's everywhere (the '23 curse') prior to enormous success and fruitfulness. 23rdians are a group of persons who subscribe to the mystical power of 23 and see it in multiple combinations throughout daily life.

The 23rd letter of the alphabet is 'W'. Therefore, white supremacists and racist skinheads use 23 in tattoos to represent 'W', as an abbreviation for the word "white".

The Earth is tilted at an angle of 23.44 degrees (at the moment!).

Buy your favorite Number (23) here.