Zoo of Numbers: Numbers 13 to 23

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.

is 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 = 120² + 121²

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

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

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:

The numerical pattern abcabc is divisible by 13.

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.

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)

Number anagram: ELEVEN + TWO = TWELVE + ONE

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

A million seconds is 13 days.

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

'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 -

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.

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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

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.

= 2 + 3 + 4 + 5
= (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!

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

Most people fart 14 times a day on average...

February 14 is St Valentine's day.

FOURTEEN in different languages
(Middle English fourtene, from Old English fêowertêne)
Indo-european | *K^WETWORDEKM- |
Sanskrit | CATURDASA |
Greek, Attic | τέτταρες καὶ δέκα TETTARES KAI DEKA | (new Greek: dekatessares)
Latin | QVATTVORDECIM |, Archaic Latin | *QVATBORDECEM |
Italian quattordici; French, Portuguese quatorze; Spanish catorce; Romanian paisprezece.
Old German | FIORZEHAN |
German vierzehn; Dutch veertien; Danish fjorten.
Russian четырнадцать chetyrnadtsat'; Slovene štirinajst; Polish czternaście.
Magyar tizennégy.
Chinese 十四 shisi.
Hebrew ארבע עשרה arba'a-'asar; Arabic اربعة عشر arba'a-taash.
Nahuatl Mahtlāctlinnāhui.

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

= 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

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

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

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.

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

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

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

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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.
is the base of the hexadecimal number system, which is used extensively in computer science.

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

16 + 9 = 25 = 5²
16 x 9 = 144 = 12²

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').

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

There are 16 orders of mammals.

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 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.

seems to be the lowest possible number of givens for a sudoku puzzle with a unique solution (this has yet to be proven).

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

= 9² - 8²
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 (Pythagorean triple)
17² = 1³ + 2³ + 4³ + 6³
17² = 0⁰ + 1¹ + 2² + 3³ + 4⁴
17³ = 4,913 and 4 + 9 + 1 + 3 = 17

_{= 2}2²_{+ 1}

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.

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

Most Italians believe that 17 is unlucky, and they are very superstitious concerning it. For instance, Air Italia has no 17th row, Italian buildings do not have a seventeenth floor, 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 culture. In fact, XVII rearranged spells VIXI, which means "I am dead" in Latin. The fear of the number 17 is called 'heptadecaphobia' or 'heptakaidekaphobia'.

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

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

is the number of syllables in a haiku.

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

A day on Uranus lasts for 17 hours.

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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)
= (2 + 8)²

18 ₌ 9 + 9
81    9 x 9

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.

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

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

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

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

is a repfigit number.
is the maximum number of 4^th powers needed to sum to any number.
With respect to divisibility, nineteen has a very simple test: 19 divides 100a + b if, and only if, it divides a + 4b.

19 = 19 = 1
95    95    5

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

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

= F + 4 (base 16)

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.

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

= 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)

≈ e - (very very close to 20!)

20 = 8 PM

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

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

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.

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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.
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
= 4⁰ + 4¹ + 4²
= 6² - 5² + 4² - 3² + 2² - 1²

2²¹ - 21 is prime.

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).

"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.

is a pentagonal number (22 = 1 + 4 + 7 + 10) and a centered heptagonal number.
is the number of partitions of 8.
22/7 = 3.142... Honest approximation of .
= κβ (in Ionic Greek numeration)

There are 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, 1+1+1+1+1+1+1+1

"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.

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!).

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