15 (number)

M = 15

The 15 perfect matchings of K₆

15 as the difference of two positive squares (in orange).

15 is:

• The eighth composite number and the sixth semiprime and the first odd and fourth discrete semiprime;[1] its proper divisors are 1, 3, and 5, so the first of the form (3.q),[2] where q is a higher prime.
• a deficient number, a smooth number, a lucky number, a bell number (i.e., the number of partitions for a set of size 4), [3] a pentatope number, [4] and a repdigit in binary (1111) and quaternary (33). In hexadecimal, and higher bases, it is represented as F.
• with an aliquot sum of 9; within an aliquot sequence of three composite numbers (15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
• the second member of the first cluster of two discrete semiprimes (14, 15); the next such cluster is (21, 22).
• a triangular number, a hexagonal number,[5] and a centered tetrahedral number.
• the number of partitions of 7.
• the smallest number that can be factorized using Shor's quantum algorithm.
• the magic constant of the unique order-3 normal magic square.
• the number of supersingular primes.
• the smallest positive number that can be expressed as the difference of two positive squares in more than one way:[6] ${\displaystyle 4^{2}-1^{2}}$ or ${\displaystyle 8^{2}-7^{2}}$ (see image).

Furthermore,

• 15's prime factors, (3 and 5) also form its main factor pair (3 x 5) and both comprise the first twin-prime pair.
• 15 is one of two numbers within the teen numerical range (13-19) not to use a single-digit number in the prefix of its name (the first syllable preceding the teen suffix); instead, it uses the adjective form of five (fifth) as the prefix. The other such number is 13.
• The first 15 superabundant numbers are the same as the first 15 colossally abundant numbers.
• In decimal, 15 contains the digits 1 and 5 and is the result of adding together the integers from 1 to 5 (1 + 2 + 3 + 4 + 5 = 15). The only other number with this property (in decimal) is 27.
• There are 15 truncatable primes that are both right-truncatable and left-truncatable:

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequence A020994 in the OEIS)

• There are 15 perfect matchings of the complete graph K₆ and 15 rooted binary trees with four labeled leaves, both of these being among the types of objects counted by double factorials.
• With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet (11, 13, 17, 19).
• If a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems.
• 15 is the product of distinct Fermat primes, 3 and 5; hence, a regular pentadecagon is constructible with a compass and unmarked straightedge, and ${\displaystyle \cos {\frac {\pi }{15}}}$ is expressible in terms of square roots.
• There are 15 monohedral convex pentagonal tilings, with eight being edge-to-edge.
• There are 15 regular and semiregular tilings when infinite (improper) apeirogonal forms are counted: three are regular (with one self-dual), eight are semiregular (with one chiral), and four are apeirogonal (from a total of 8, in-which 4 are duplicates).
• Full icosahedral symmetry contains 15 mirror planes (2-fold axes). Specifically, the symmetry order for both the regular icosahedron and regular dodecahedron (which is made of regular pentagons) is 120: equal to sum of the first 15 integers, and the factorial of 5, wherein the sum of the first 5 integers itself is 15. Expressed mathematically:

  ${\displaystyle \sum _{i=1}^{15}i=120}$, while ${\displaystyle \sum _{i=1}^{5}i=15}$, and ${\displaystyle 5!=120}$.

• There are 15 Archimedean solids and 15 Catalan solids when enantiomorphic forms are counted separately.
• There are 15 regular honeycombs in hyperbolic 3-space: four are compact, and 11 are paracompact.[7]
• It is the smallest non-trivial Surprise Number. When 15 is partitioned into 2 parts of digits- smaller part 1 and larger part 5, adding numbers from smaller 1 to larger 5 = 1 + 2 + 3 + 4 + 5 = 15 = Original Number.

Other such numbers include 0, 1, 27, 190, 204, 216, 429, 1353, 1863, 3078, 3388, 3591, 7119, 19900, 20328, 21252, 21762, 23287, 23490, 78403, 133533, 178623, 1301613, 1999000, 2002077, 2006118, 2077402, 2132532, 2177622, 2282148, 2732353, 3882813, etc.

Seashells from the mollusk Donax variabilis have 15 coloring pattern phenotypes.

• The atomic number of phosphorus.
• Group 15 of the periodic table are sometimes known as the pnictogens.
• 15 Eunomia is the largest Eunomian asteroid in the inner asteroid belt.

15 is designated as an emergency number in Pakistan, for mobile phones, similar to the international GSM emergency number 112;[9] if 112 is used in Pakistan, then the call is routed to 15. 112 can be used in an emergency even if the phone is locked and does not have a SIM card in it.

• In all four of the major Gaelic games—hurling, Gaelic football for men and women, and camogie—each team has 15 players on the field at any given time.
• In both major forms of gridiron football (American and Canadian), standard games are played in 15-minute quarters.
• In rugby union:
  • Each team has 15 players on the field at any given time, unless one or more players are removed for a serious rules infraction.
  • The jersey number 15 is worn by the starting fullback.
• In tennis, the number 15 represents the first point gained in a game.
• In NCAA Division I women's college basketball, teams are allowed to provide scholarships to no more than 15 players at any given time.
• In Australian rules football, games in the AFL Women's league consist of 15-minute quarters.
• The maximum number of players on an NBA team roster.
• On 17 December 2022, the Sydney Thunder were bowled out for just 15 runs in a loss to the Adelaide Strikers during the 2022–23 Big Bash League season. It is the lowest score ever recorded by any professional team in worldwide Twenty20 cricket history.[10]

• The age of a quinceañera, a Latina girl celebrating her 15th birthday. Thus, Spanish bingo callers might refer to the number 15 as la niña bonita (the beautiful girl).
• The age for obtaining a driver's (or learner's) permit in certain jurisdictions, including some, where the legal age for a driver's license is 16.
• The minimum age for driving a moped in some countries.
• The age of criminal responsibility in various countries.
• In the UK, a minor can be sent to prison to await trial at the age of 15.
• In some countries, it is the legal age for sexual consent.[11]
• In the UK, it is the legal age to watch a BBFC "15" rated film in a cinema or purchase a "15" rated film or video game.
• The drinking age in Ethiopia.
• The age of majority in Indonesia, Myanmar, Iran and Yemen.

• "Fifteen" is a song composed by Taylor Swift on her album Fearless

Fifteen is:

• The 15 puzzle.
• The number of minutes in one quarter of an hour; 15 minutes past or before an hour is often known as quarter past and quarter to, respectively.

• List of highways numbered 15

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 91–93

Look up 15 in Wiktionary, the free dictionary.

• Clewett, James. "15: f in hexadecimal". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-02. – discussing hexadecimals
• Bowley, Roger. "15: Bumfit". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-01. – discussing the Celtic number as used in Lincolnshire