Five is conjectured to be the only odd untouchable number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.
Five is also the only prime that is the sum of two consecutive primes, namely 2 and 3.
The number 5 is the fifth Fibonacci number, being 2 plus 3. It is the only Fibonacci number that is equal to its position. 5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers.
Five is the second Sierpinski number of the first kind, and can be written as S2 = (2^{2}) + 1.
While polynomial equations of degree 4 and below can be solved with radicals, equations of degree 5 and higher cannot generally be so solved. This is the Abel–Ruffini theorem. This is related to the fact that the symmetric groupS_{n} is a solvable group for n ≤ 4 and not solvable for n ≥ 5.
While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar: K_{5}, the complete graph with 5 vertices.
Five is the only prime number to end in the digit 5 because all other numbers written with a 5 in the ones place under the decimal system are multiples of five. As a consequence of this, 5 is in base 10 a 1-automorphic number.
Vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the decimal system, all multiples of 5 will end in either 5 or 0.
In the powers of 5, every power ends with the number five and from 5^{3}, if the exponent is odd, then the hundreds digit is 1; instead, if it is even, the hundreds digit is 6.
In fifth powers, n^{5} ends in the same digit as n.
Evolution of the glyph
The evolution of the modern Western glyph for the numeral 5 cannot be traced back to the Indian system as for the numbers 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several different glyphs which bear no resemblance to the modern glyph. The Nagari and Punjabi took these glyphs and all came up with glyphs that are similar to a lowercase "h" rotated 180°. The Ghubar Arabs transformed the glyph in several different ways, producing glyphs that were more similar to the numbers 4 or 3 than to the number 5.^{[4]} It was from those characters that Europeans finally came up with the modern 5.
While the shape of the 5 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .
In Discordianism, 5 is seen as a very important number. This is demonstrated in the Law of Fives, as well as in the Pentabarf, which contains five rules.
Each page of the Principia Discordia—the primary religious document in Discordianism—is labeled with five digits.
According to Shia Muslims, the Panjetan or the Five Holy Purified Ones are the members of Muhammad's family: Muhammad, Ali, Fatimah, Hasan, and Husayn and is often symbolically represented by an image of the Khamsa.
Judaism
The Torah contains five books—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—which are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (חומש, Hebrew for "fifth").
The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.
Sikhism
The five sacred Sikh symbols prescribed by Guru Gobind Singh are commonly known as panj kakars or the "Five Ks" because they start with letter K representing kakka (ਕ) in the Punjabi language's Gurmukhi script. They are: kesh (unshorn hair), kangha (the comb), kara (the steel bracelet), kachhehra (the soldiers shorts), and kirpan (the sword) (in Gurmukhi: ਕੇਸ, ਕੰਘਾ, ਕੜਾ, ਕਛਹਰਾ, ਕਿਰਪਾਨ). Also, there are five deadly evils: kam (lust), krodh (anger), moh (attachment), lobh (greed), and ankhar (ego).
In Cantonese, "five" sounds like the word "not" (character: 唔). When five appears in front of a lucky number, e.g. "58", the result is considered unlucky.
In East Asian tradition, there are five elements: (water, fire, earth, wood, and metal). The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye. Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday.
Members of The Nation of Gods and Earths, a primarily African American religious organization, call themselves the "Five-Percenters" because they believe that only 5% of mankind is truly enlightened.
The Mach Five (マッハ号 Mahha-gō?), the racing car Speed Racer (Go Mifune in the Japanese version) drives in the anime series of the same name (known as "Mach Go! Go! Go!" in Japan)
Myst uses the number 5 as a unique base counting system. In The Myst Reader series, it is further explained that the number 5 is considered a holy number in the fictional D'ni society.
Number Five is also a character in The Umbrella Academy comic book and TV series adaptation
V for Vendetta (2005), produced by Warner Bros., directed by James McTeigue, and adapted from Alan Moore's graphic novel V for Vendetta prominently features number 5 and Roman Numeral V; the story is based on the historical event in which a group of men attempted to destroy Parliament on November 5, 1605
Ben Folds Five, an American alternative rock trio, 1993–2000, 2008 and 2011–2013
R5 (band), an American pop and alternative rock group, 2009-2018
Other uses
A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.
Modern musical notation uses a musical staff made of five horizontal lines.
In harmonics – the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A majortriadchord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.
Odyssey 5, a 2002 science fiction television series
Tillbaka till Vintergatan, a Swedish children's television series featuring a character named "Femman" (meaning five), only can utter the word 'five'.
The Five(talk show): Fox News Channel roundtable current events television show, premiered 2011. So named for its panel of five commentators.
Yes! PreCure 5 is a 2007 anime series which follows the adventures of Nozomi and her friends. It is also followed by the 2008 sequel Yes! Pretty Cure 5 GoGo!
The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).
The number 5 is used to represent the position of center.
Each team has five players on the court at a given time. Thus, the phrase "five on five" is commonly used to describe standard competitive basketball.
The "5-second rule" refers to several related rules designed to promote continuous play. In all cases, violation of the rule results in a turnover.
Under the FIBA (used for all international play, and most non-US leagues) and NCAA women's rule sets, a team begins shooting bonus free throws once its opponent has committed five personal fouls in a quarter.
There are five different ways that a player can score a goal (teams at even strength, team on the power play, team playing shorthanded, penalty shot, and empty net).
The area between the goaltender's legs is known as the five-hole.
In most rugby league competitions, the starting left wing wears this number. An exception is the Super League, which uses static squad numbering.
One of the two starting lock forwards wears number 5, and usually jumps at number 4 in the line-out.
In the French variation of the bonus points system, a bonus point in the league standings is awarded to a team that loses by 5 or fewer points.
Technology
5 is the most common number of gears for automobiles with manual transmission.
In radio communication, the term "Five by five" is used to indicate perfect signal strength and clarity.
On almost all devices with a numeric keypad such as telephones, computers, etc., the 5 key has a raised dot or raised bar to make dialing easier. Persons who are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers can be found with their relative position around the 5 button (on computer keyboards, the 5 key of the numpad has the raised dot or bar, but the 5 key that shifts with % does not).
On most telephones, the 5 key is associated with the letters J, K, and L, but on some of the BlackBerry phones, it is the key for G and H.
In the computer game Riven, 5 is considered a holy number, and is a recurring theme throughout the game, appearing in hundreds of places, from the number of islands in the game to the number of bolts on pieces of machinery.
^Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
^Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.65
^Kisia, S. M. (2010), Vertebrates: Structures and Functions, Biological Systems in Vertebrates, CRC Press, p. 106, ISBN978-1-4398-4052-8, The typical limb of tetrapods is the pentadactyl limb (Gr. penta, five) that has five toes. Tetrapods evolved from an ancestor that had limbs with five toes. ... Even though the number of digits in different vertebrates may vary from five, vertebrates develop from an embryonic five-digit stage.