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 the smallest number of integer-sided rectangles that tile a rectangle so that no 2 rectangles share a common length.
is a Mersenne prime (2³ - 1).
is the unity digit of the number 3³¹.
is the positive solution to: 3x² = 16x + 35
is probably the smallest prime whose 4^th power is of the form a⁴+b⁴-c⁴:
7⁴ = 157⁴ + 227⁴ - 239⁴
- posted by Morten Risager

= 4 + 3 = 4² - 3²
= 1² + 1² + 1² + 2² (sum of 4 squares)
= 2⁵ - 5²

1⁷ + 4⁷ + 4⁷ + 5⁷ + 9⁷ + 9⁷ + 2⁷ + 9⁷ = 14459929

7¹ + 1 = 2 x 2²
7² + 1 = 2 x 5² (this property is used in the magic Tangram)

7² = 3⁴ - 2⁵
7³ = 2⁹ - 13²

Is the only solution to the following equations:
x = 2a² - 1
and x² = 2b² - 1
(x = 7, a = 2, and b = 5)

No perfect square ends in a 7 (perfect squares can only end in a 0, 1, 4, 5, 6, or 9).

The only single digit prime number to produce a square when its cube and the cube's proper divisors are added:
7¹ + 7² + 7³ + 1 = 20²

Some multiples of 7 are palindromic, such as:
77, 161, 252, 343,434, 525, 595, 616, 686, 707, 777, 868, 959, 1001, 1771, 2002, 2772, 3003, 3773, 4004, 4774, 5005, ...

The smallest positive integer whose reciprocal (1/7) has a pattern of more than one repeating digit:
1/7 = 0.142857142857142857...
It is also the smallest number for which the periodic sequence of 1/n is of length n-1. The next such numbers are 17, 19, 23, 29, 47, 59, 61, 97, ...

Other notable properties:
by multiplying the cyclic number 142857 by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself...
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142
- posted by Larry Bickford -

Interestingly, if we square the last three digits of 142857 and subtract the square of the first three digits, we also get back a cyclic permutation of the number!
857² - 142² = 714285

999,999 / 7 = 142857

Adding 3 to the reciprocal of 7 gives a rough approximation of Pi (3.142857...).

(987'654'321 - (123'456'789 + 9)) / 7 = 123'456'789
- by Gianni A. Sarcone in 'Il Mio Computer', nr. 101 -

Divisibility by Seven: take the digits of the number in reverse order, from right to left, multiplying them successively by the digits 1, 3, 2, 6, 4, 5, ... (repeating with this sequence of 6 multipliers as long as necessary). Add the products.
For example: is 1708 divisible by 7? Well, reverse the number, 8071, multiply and add its digits as explained, 8x1+0x3+7x2+1x6 = 28. The sum is divisible by 7, so 1708 is.

Divisibility by Seven (2): for numbers having 3 digits...
154 > (2)+1+5+4+(2) = 14 is divisible by 7, so 154 is.
245 > (4)+2+4+5+(6) = 21 is divisible by 7, so 245 is.
862 > (2)+8+6+2+(1) = 19 and 862 are not multiple of 7.
366 > (6)+3+6+6+(3) = 24 and 366 are not multiple of 7.
The numbers in brackets are added in order to form with their neighbors a 2-digit number multiple of 7. If the sum of all digits is a multiple of 7, then the related number is divisible by 7.
- posted by Silvio Moura Velho -

is a prime number
7 is a prime number
73 is a prime number
739 is a prime number
7393 is a prime number
73939 is a prime number
739391 is a prime number
7393913 is a prime number
73939133 is a prime number
(73939133 is the largest right-truncatable prime)

The opposite sides of a die cube always add up to 7

A square piece of paper cannot be folded in half more than 7 times... However, Britney Gallivan (of Pomona, California) found a formula that tells us to successfully fold paper 12 times, we would need about 1.2 km of paper... and she did it! - posted by Chris GIbbons

The Bolans of West Africa used for reckoning a kind of heptal (base-7) positional notation system.

Curiously, 'letters' contains seven times itself (selfreferential word). - G. Sarcone

In Japan, the word shi means 'death'. In some cases nana ('7' in early classical Japanese) replaces the word shichi ('7', in modern Japanese) because it contains shi.

The symbol of the Judaism Faith is the menorah, a 7-branch candlestick (see opposite).

"Qixing" (七個星), in the Mystical Numbers of Taoism, represents the number 7 and symbolizes the Seven Stars -- The Big Dipper.

"Sju sjösjuka sjömän på det sjunkande skeppet Shanghai" (seven seasick seamen on the sinking ship Shanghai), is a Swedish tongue twister.

7 ± 2 is a cognitive numeric limit (cf. 'The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information' by the cognitive psychologist George A. Miller, The Psychological Review, 1956, vol. 63, pp. 81-97)

Even the formula of the Coca-Cola soft-drink is related to number 7. In fact, Merchandise 7X is the "secret ingredient" in Coca-Cola and has apparently remained a secret since its invention in 1886.

MAGIC SEVEN
7 is the Holy number. There are 7 days in creation, 7 days in the week, 7 phases of the moon, every 7th year was sabbatical, and 7 times 7 years was the jubilee. There are 7 ages in the life of man, 7 divisions in the Lord's Prayer, 7 bibles, 7 churches of Asia, 7 Graces, 7 Deadly Sins, 7 Senses, 7 Sorrows of the Virgin, 7 Virtues, 7 joys of the Virgin, 7 Precious Things of the Buddhas, 7 Sleepers of Ephesus, 7 Lamps of Architecture. The apostles chose 7 deacons, Enoch, who was translated, was 7th from Adam; Jesus Christ was the 77th in a direct line. The vision of Daniel was 70 weeks; and the elders of Israel were 70. There were also 7 heavens, 7 planets, 7 stars, 7 wise men, 7 champions of Christendom, 7 notes in music, 7 primary colors, 7 sacraments of the Catholic Church, and 7 wonders of the world (see below). 7 gateways were traversed by Inanna, the most important goddess of the Sumerian pantheon, during her descent into the underworld. The 7th son was considered endowed with pre-eminent wisdom; and the 7th son of a 7th son is still thought to possess the power of healing diseases spontaneously.

The 7 wonders of the ancient world are:
- Great Pyramid of Giza,
- Hanging Gardens of Babylon,
- Temple of Artemis at Ephesus,
- Statue of Zeus at Olympia,
- Mausoleum of Maussollos at Halicarnassus,
- Colossus of Rhodes,
- Lighthouse of Alexandria.

The 'Seven Blunders of the World' according to Mahatma Gandhi are:
- Wealth without work,
- Pleasure without conscience,
- Knowledge without character,
- Commerce without morality,
- Science without humanity,
- Worship without sacrifice,
- Politics without principle.

SEVEN in different languages
(Old English seofon)
Indo-european | *SEPTM |
Sanskrit | SAPTA |
Greek, Attic | 'EΠTA HEPTA |
Latin | SEPTEM |
Italian sette; French sept; Spanish siete; Catalan and Provençal set; Portuguese setes; Romanian şapte.
Old Germanic | SIBUM |
Dutch zeven; German sieben; Danish syv; Norvegian sju.
Old Slavic | SEDMI |
Russian семь syem'; Czech sedm.
Chinese 七 qī.
Semitic root | SH-B-^c |
Ancient Egyptian sefekh; Akkadian sebe;
Arabic سبعة sab'a; Hebrew שבעה sheh-vah.
The roots of the word seven are hidden in the following words: hebdomadal, heptagon, heptahedron, september, septennial, septentrion, septuple; Bathsheba, Elizabeth. In French: semaine, septuor.

Buy your favorite Number (7) here.

is the second honest number, because 8 = "two cubed".
One can split a pie into 8 pieces with three straight cuts.
is the largest cube (2³) in the Fibonacci sequence.

8 = 5 + 1 + 2 and 512 = 8³

(2n - 1)² ≡ 1 (mod 8)
(... any squared odd number is a multiple of 8 plus 1)

Digital sums of prime pair products:
5 x 7 = 35 and 3+5 = 8
11 x 13 = 143 and 1+4+3 = 8
17 x 19 = 323 and 3+2+3 = 8
29 x 31 = 899 and 8+9+9 = 26 and 2+6 = 8
41 x 43 = 1763 and 1+7+6+3 = 17 and 1+7 = 8
59 x 61 = 3599 and 3+5+9+9 = 26 and 2+6 = 8 ... etc.

A base-8 (octal) numeral system was devised by the Yuki of Northern California, who used the spaces between the fingers to count.

Zenzizenzizenzic was a term coined by earlier mathematicians to indicate the 8 power of a number. The root word is the German zenzic from the medieval Italian word censo. The Italians used censo to translate the Arabic word mál (literally "possessions, property") meaning 'squared' in the context of mathematics.

1 human being out of 8 is a Chinese farmer.
Inhabitants of the world (2004): 6.2 billions.
Inhabitants of China (2004): 1.3 billions, 60% of them living in a rural area.
The proportion is then: (1.3 · 60%)/6.2 = approx. 1 Chinese farmer / 8 human beings.
--- Is this article still topical? Please send us your comments!

In numerology, 8 is the number of building, and in some theories, also the number of destruction.

"Bagua", in the Mystical Numbers of Taoism, represents the number 8 and symbolizes the Eight Signs of Change and Combinations.

Eight is considered a lucky number in Chinese culture because it sounds like the word "prosper". In Northern China, people often say: "If you want to be rich you must stick to the number eight". Dates and times with either a 6 or an 8 have been predominantly chosen as auspicious for important events. When Chinese businessmen stay in a hotel, they would rather choose rooms with numbers such as: 6, 8, 58, 168, 518, 588, 868, and so on.

Hanukkah is an eight day Jewish holiday that starts on the 25th day of December.

Joke. What did 0 say to 8? Answer: "Nice belt!".

Buy your favorite Number (8) here.

is the largest number of cubes needed to sum to any positive integer.

= 5 + 4 = 5² - 4²
= 3²
= 1³ + 2³
= !4 = 4! (1-1/1!+1/2!-1/3!+1/4!)

2⁵ x 9² = 2592

0 1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1 0 is a mnemonic trick to remember your 9x table.

(9 x 123'456'789) - 123'456'789 = 987'654'321 - 9

= 97524/10836
= 95823/10647
= 95742/10638
= 75249/08361
= 58239/06471
= 57429/06381
(in each fraction above all digits from 0 to 9 were used once)

Curious arrangements with 9's:
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1 111
1234 x 9 + 5 = 11 111
12345 x 9 + 6 = 111 111
123456 x 9 + 7 = 1 111 111
1234567 x 9 + 8 = 11 111 111
12345678 x 9 + 9 = 111 111 111
The results above are all triangular numbers in base-9.

1/9 = 1/10 + 1/10² + 1/10³ + 1/10⁴ + 1/10⁵ + ... = ∑_i=1 10^-i

Notably, the recurrent number 0.999999... denotes a real number which is equal to 1:
9 x 1/9 = 9 x 0.111111...
Thus, 1 = 0.999999...

₉9⁹_{= 1.9662705 x 10}77

_{but 9}(9⁹) _{= a number containing 369,000,000 of digits!}

To multiply 10,112,359,550,561,797,752,808,988,764,044,943,820,224,719 by 9 you just have to move the 9 at the very end up to the front. Apparently, it is the only number that does this!

1/1089 = 0,00091827364554637281... (the decimal expansion is a sequence of numbers of the 9 times table: 9, 18, 27, 36, ...)

Water expands by about 9% as it freezes.

9.81 meters per second squared is g, the gravity constant on earth.

In probability, the 'nine' is a logarithmic measure of reliability or probability of an event, defined as the negative of the base-10 logarithm of the probability of the event's complement. For instance, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to -log₁₀ 0.01 = 2 nines of probability.

09:F9:11:02:9D:74:E3:5B:D8:41:56:C5:63:56:88:C0... This string of hexadecimal numbers appears to be the HD-DVD processing key useful to decrypt and play most HD-DVD movies in Linux.

NUNDINAE ("9th day", the Roman market day): from earliest times, a market day occurred every eight days - every 9th day, according to the Roman method of inclusive reckoning. It was a day of rest from agricultural labor and a time to take produce to market.

Being the number of fingers or toes, 10 became the base of the decimal number system. It is the number of completeness or finality. With ten as complete, 9 comes into prominence as almost complete. Troy was besieged for 9 days and fell on the tenth. Odysseus (Lat. Ulysses) wandered for 9 years and arrived home on the tenth.

In Guangdong province, China, nine is associated with eternity or power.

"Jiugong" (also known as "Jiuxuan"), in the Mystical Numbers of Taoism, represents the number 9 and symbolizes the Nine Chambers. It is the last number of order before returning back to chaos. After that, the cycle repeats...

The symbol of the Bahai Faith is a 9-pointed star (see opposite).

A full moon is nine times brighter than a half moon.

The standard dimensions of a play card are: 6.35 x 9 cm.

The first nine letters of the Maldivian alphabet (h, sh, n, r, b, lh, k, ', and v) are graphically derived from the Arabic numerals from 1 to 9: ١, ٢, ٣, ٤, ٥, ٦, ٧, ٨, ٩.

is the base of our numerical system.
is a triangular number: 1 + 2 + 3 + 4 = 10
is the only triangular number which is also the sum of 2 consecutive square odd numbers: 1² + 3² = 10
cannot be the difference of 2 squares, because 10 is of the form 4n + 2

= (5 + -15) + (5 - -15) = (5 + -15)(5 - -15)/4

10! = 6! x 7!

= 2 + 3 + 5 (sum of the first 3 primes)
= 1 / (1 - (3 / (3 + 1/3)))
= ^googolgoogolplex

There seems to be only 11 powers of ten that are products of two integers without any zero digits:

Some people argue that we only use 10% of our brains... No, this is wrong: while the majority of the brain may not be active at any one moment, all of it is essential for normal function. See mythical numbers.

In most advertisements, including newspapers, the time displayed on a watch is 10:10.
The reason for this is to "frame" the logo of the watch maker. The hands at that number cause the eyes to look right at the logo! - posted by Shaun Hussey, Graphic Designer
The hands on the dial indicating 10:10 also represent a smiling face, sells better than watches indicating e.g. 08:20. - posted by Pander
The actual reason for this is that during the second world war, Japan was the main manufacturer of clocks and watches, and when the US bombed Hiroshima islands, the impact of the explosion was so much that all the clocks and watches stopped at that particular time, and that time was 10 minutes past 10. So, in order to mark the gruesome bombings and remember the same, all over the world the clocks and watches at any shop show this time. - Posted by Saurabh Gulati

"One father is better at caring for ten children than ten children are for one father."

is the third honest number, because 11 = "two plus nine".
is the only palindromic prime with an even number of digits.
is the smallest prime with multiplicative and additive persistence of 1.

Any power of 11 ends in a 1.

11¹¹ contains two embedded elevens.

Any large number is divisible by 11 if the difference between the sum of its odd digits (units, hundreds, etc.) and the sum of its even digits (tens, thousands, etc.) is 0 or a number divisible by 11.

Displayed on a calculator, it reads the same whether the calculator is turned upside down or reflected on a mirror, or both.

'Eleven plus two' is the anagram of 'twelve plus one'.

The Maoris, the initial inhabitants of New Zealand, used for reckoning or accounting purposes the undecimal (base-11) positional notation system.

11² = 3⁰ + 3¹ + 3² + 3³ + 3⁴
11³ = 3² + 19² + 31²
11 x 11 = 65 + 56 (palindromic equality)
11² x 9182736455463728191 = 1111111111111111111111
- posted by Yosuke Ikeda, Yasuji Kondo, Yasuhiko Sakaitani, Ken Hirotomi, Hiroyuki Ozaki, Atushi Tanaka, Ryohei Miyadera (Kwansei Gakuin High School) -

11 + 1.1 = 11 x 1.1

Magic triangle with a constant of 11;
2 + 3 + 6 = 2 + 5 + 4 = 6 + 1 + 4 = 11

11:11 Phenomenon
If your attention has been drawn to clocks and watches at exactly 11:11, 12:12, 10:10, 22:22, 12:34, 2:22, 3:33, 4:44 or 5:55, then you are a 11:11-phenomenon addict... In fact, some people believe that when they notice the number pattern 1111 or the digits 11:11 on digital time pieces, time/temperature signs, and so on, that there is an underlying reason besides mere chance or coincidence. A belief of this nature is related to the idea of synchronicity.

11 Dimensions
According to Dr Michio Kaku our universe is a four-dimensional ball floating in eleven dimensions.

is a pentagonal number.
is the smallest number with exactly six divisors, and the smallest abundant number.
is the largest known even number expressible as the sum of two primes in one way (5 + 7).

= 22 / (1 + (1 / (1 + 1/5)))
= (0! + 0! + 0!)! x (0! + 0!)
= (-3i⁵)(4i)

= 2⁴ - 2²
= 3¹ + 3²
= 4² - 4¹

12² x 1122334455667789 = 161616161616161616 (periodic number)
- posted by Yosuke Ikeda, Yasuji Kondo, Yasuhiko Sakaitani, Ken Hirotomi, Hiroyuki Ozaki, Atushi Tanaka, Ryohei Miyadera (Kwansei Gakuin High School) -

12 = 7³ + 10³ - 11³
12² = 10³ - 8³ - 7³ - 1³
12³ = 9³ + 10³ - 1³
12⁵ = 4⁵ + 5⁵ + 6⁵ + 7⁵ + 9⁵ + 11⁵

1/12² = 1/15² + 1/20² (smallest integers of the reciprocal pythagorean equation)

12 x 12 = 144 is palindromic to 441 = 21 x 21
12 x 12 + 21 x 21 = 585 is also palindromic
other curious palindromic equalities involving 12:
12 x 42 = 24 x 21
12 x 63 = 36 x 21
12 x 84 = 48 x 21

12 + 21 = 3 + 4 + 5 + 6 + 7 + 8

12! + 34567 is prime.

12 x 3² + 3 = 111
400/33 = 12.1212121212...

There are 12 primary ways of arranging 8 queens on a chessboard so that no queen can capture any other queen.

Two + eleven = twoeleven, twOelEveN - ONE = twelve.

are the symbols of the phone dial.
Telephone Number Mnemonics
American visitors are familiar with an ingenious method of making telephone numbers more memorable. In the US many companies advertise their telephone numbers with words. For example "Call 1800-DOG-HELP" might be the advertising slogan for a dog rescue center. This is much easier to remember than "Call 1800-364-4357".
To aid this system, the buttons on telephones are marked with letters, as shown on the right. This method has been used in Britain too, although it is now far less popular.

Which is correct: 12 Midnight A.M. or 12 Midnight P.M.? Neither!
12 midnight A.M. and 12 midnight P.M., or 00:00 A.M. and 00:00 P.M., mean nothing at all. They are simply the midpoints that divide the day into two equal halves.
Each and every day begins exactly at midnight, and each A.M. begins precisely thereafter. Similarly, each P.M. begins immediately after noon. No meaning can be assigned to 12:00 A.M. (00:00 A.M.), or to 12:00 P.M. (00:00 P.M.). They are merely reference points meant to simplify timetables for us.

A dozen is a group or set of 12: a dozen bottles of sherry.

A shock is a group of 12 sheaves of grain stacked upright in a field for drying. The word shock means also a lot consisting of 60 pieces; -- a term applied in some Baltic ports to loose goods.