Fact-checked by Grok 2 weeks ago

21

Twenty-one (21) is the natural number following 20 and preceding 22.^[1] It is the product of the prime numbers 3 and 7, making it a semiprime with distinct prime factors and a square-free integer.^[1] In number theory, 21 belongs to several notable sequences, including the Fibonacci sequence as its ninth term (where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21) and the triangular numbers as the sixth term (the sum of the first six natural numbers: 1 + 2 + 3 + 4 + 5 + 6 = 21).^[2] Additionally, 21 is a Harshad number, divisible by the sum of its digits (2 + 1 = 3, and 21 ÷ 3 = 7), and the smallest number requiring 21 distinct squares to tile a larger square.^[3]^[4] Beyond mathematics, 21 holds cultural and historical significance in various domains. In military tradition, the 21-gun salute represents the highest honor, originating from naval customs where ships fired 21 shots to demonstrate peaceful intentions, with the number derived from multiples of the sacred 7 in several cultures (7 × 3 = 21).^[5] Historically, 21 was the age of majority in the United States and many other jurisdictions, granting full legal adulthood for purposes like voting, contracting, and jury service until the 26th Amendment in 1971 lowered the voting age to 18, prompting similar changes elsewhere.^[6]^[7] In gaming, blackjack, also known as 21, is a popular casino card game where players aim to achieve a hand value closest to 21 without exceeding it, drawing from 17th-century European variants like vingt-et-un.^[8] The number 21 has also inspired media and popular culture. The 2008 film 21, directed by Robert Luketic, dramatizes the true story of MIT students who used card-counting techniques to win millions at blackjack in Las Vegas during the 1990s, based on the book *Bringing Down the House* by Ben Mezrich.^[9] In sports and other contexts, 21 often symbolizes completion or milestone, such as the 21 stages of the Tour de France or the gestation period of about 21 days for certain small mammals, such as meadow voles or rats, in veterinary references, though its mathematical and ceremonial roles remain most defining.

Mathematics

Numerical properties

Twenty-one (21) is the natural number following 20 and preceding 22.^[10] It factors as the product of the distinct primes 3 and 7, making it a square-free semiprime.^[11] Among square-free semiprimes, ordered by magnitude, 21 is the fifth, following 6, 10, 14, and 15.^[12] It is the second semiprime of the form $3 \times q, where q is a prime greater than 3 (after $3 \times 5 = 15).^[11] The positive divisors of 21 are 1, 3, 7, and 21.^[13] The sum of these divisors is \sigma(21) = 32. The proper divisors, excluding 21 itself, are 1, 3, and 7. In base 4, 21 has the repdigit representation $111_4, as

$1 \times 4^2 + 1 \times 4^1 + 1 \times 4^0 = 16 + 4 + 1 = 21.

^[14] Twenty-one is the smallest Blum integer, defined as the product of two distinct primes both congruent to 3 modulo 4 (here, 3 ≡ 3 mod 4 and 7 ≡ 3 mod 4).^[15] The value of Euler's totient function at 21 is \phi(21) = 12, signifying that 12 positive integers up to 21 are coprime to 21 (specifically, 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20).

Mathematical appearances

The number 21 appears prominently in various mathematical sequences and geometric constructions. As the sixth triangular number, it is given by the formula T_n = \frac{n(n+1)}{2}, so T_6 = \frac{6 \times 7}{2} = 21. Geometrically, this represents the number of objects in an equilateral triangle arranged with six objects along each side, formed by stacking rows of 1, 2, 3, 4, 5, and 6 objects.^[16] In polygonal number sequences, 21 is the third octagonal number, calculated as O_n = n(3n - 2), yielding O_3 = 3(9 - 2) = 21; it is often regarded as the first non-trivial octagonal number beyond the initial terms. Octagonal numbers correspond to the number of dots in an octagon with n dots along each side.^[17] Among linear recurrence sequences, 21 is the eighth Fibonacci number in the sequence defined by F_1 = 1, F_2 = 1, and F_n = F_{n-1} + F_{n-2} for n > 2, appearing as 1, 1, 2, 3, 5, 8, 13, 21. It is also the fifth Motzkin number, M_5 = 21, which counts the number of paths from (0,0) to (5,0) using steps (1,1), (1,-1), and (1,0) that do not go below the x-axis; these paths model lattice paths and Dyck words with level steps. Additionally, 21 is the twelfth Padovan number in the sequence defined by P_0 = 1, P_1 = 1, P_2 = 1, and P_n = P_{n-2} + P_{n-3} for n > 2.^[2]^[18]^[19] In the context of prime numbers, there are exactly 21 primes between 100 and 200, namely 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, and 199; this follows from the prime-counting function \pi(200) - \pi(100) = 46 - 25 = 21. Similarly, there are 21 two-digit prime numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. A notable geometric appearance of 21 arises in the squaring the square problem, where the smallest simple perfect squared square—a dissection of a square into 21 smaller squares of distinct sizes with no gaps or overlaps—has order 21; this unique configuration was discovered in 1978.^[20]

Science

Chemistry

In the periodic table, atomic number 21 corresponds to the chemical element scandium (Sc), classified as a rare earth metal.^[21] It was discovered in 1879 by Swedish chemist Lars Fredrik Nilson while analyzing an ytterbium sample from a Scandinavian mineral.^[21] Scandium exhibits the electron configuration [Ar] 3d¹ 4s².^[22] The metal has a silvery-white appearance that develops a slight yellowish or pinkish tint upon oxidation in air.^[23] Its physical properties include a low density of 2.985 g/cm³ and a high melting point of 1541 °C, contributing to its utility in specialized applications.^[23] Scandium has only one stable isotope, ⁴⁵Sc, which accounts for 100% of its natural abundance.^[24] Among its radioactive isotopes, ⁴³Sc and ⁴⁴Sc are notable for use in medical imaging, particularly in positron emission tomography (PET) for diagnostic purposes due to their positron emission and half-lives of approximately 3.9 hours and 4.0 hours, respectively.^[25] Scandium occurs naturally in trace amounts in various minerals, with thortveitite (Sc₂Si₂O₇) serving as its primary concentrated source, primarily found in Scandinavia and Madagascar.^[26] In practical applications, scandium is alloyed with aluminum for aerospace components, where Sc-Al alloys enhance tensile strength by up to 50% compared to conventional aluminum alloys, enabling lighter and more durable structures.^[27]

Physics

The 21 cm hydrogen line arises from the hyperfine spin-flip transition in neutral hydrogen atoms, where the spins of the proton and electron align parallel or antiparallel, emitting electromagnetic radiation at a rest frequency of 1420.405752 MHz. This corresponds to a wavelength of precisely 21.106 cm, calculated using the formula

\lambda = \frac{c}{f},

where c \approx 3 \times 10^8 m/s is the speed of light in vacuum and f is the transition frequency. The transition is highly forbidden, with a spontaneous emission lifetime on the order of 10 million years, making the line a sensitive tracer of neutral hydrogen density and velocity fields. In radio astronomy, the 21 cm line is essential for mapping the distribution of interstellar neutral hydrogen (HI), as its long wavelength penetrates dust clouds that obscure optical observations, revealing the spiral structure and kinematics of the Milky Way and other galaxies.^[28] Observations at this frequency, such as those from the Arecibo Observatory and the Very Large Array, have provided detailed maps of galactic HI, enabling studies of star formation regions and galactic rotation curves without interference from dust extinction.^[28] In cosmology, the redshifted 21 cm radiation from primordial neutral hydrogen serves as a probe of the early universe, particularly during the cosmic dawn and the epoch of reionization (EoR), when the first stars and galaxies ionized the intergalactic medium around redshifts z \approx 6-30.^[29] Detecting fluctuations in this signal allows reconstruction of the neutral hydrogen fraction, offering insights into the timing and sources of reionization, as well as the formation of the first luminous structures.^[29] Ongoing experiments, such as the Low-Frequency Array (LOFAR) and the Murchison Widefield Array (MWA), target this faint signal amid foreground contamination to constrain models of early galaxy formation.^[29]

Games and recreation

Card games

Blackjack, also known as 21, is a comparing card game between a player and a dealer, with the objective of obtaining a hand value as close to 21 as possible without exceeding it, thereby beating the dealer's hand.^[30] The game uses one or more standard 52-card decks, where numbered cards are worth their pip value, face cards (jack, queen, king) are valued at 10, and aces can count as either 1 or 11 at the player's discretion to optimize the hand total.^[31] A natural blackjack—an initial hand of an ace and a 10-value card—pays 3:2 and beats any other 21-point hand.^[31] Players receive two cards face up, while the dealer gets one face-up card and one face-down hole card. Basic player options include hitting (taking another card to increase the hand value), standing (keeping the current hand), doubling down (doubling the bet and receiving exactly one more card, typically on initial totals of 9, 10, or 11), and splitting (dividing a pair into two separate hands, with an additional bet equal to the original).^[30]^[31] Surrender, allowing the player to forfeit half the bet and end the hand early, is available in some variants. The dealer must hit on any total of 16 or less and stand on 17 or more, including soft 17 (ace counted as 11) in many rulesets; if the dealer busts (exceeds 21), all remaining players win even money.^[30]^[31] With optimal basic strategy— a set of mathematically determined decisions based on the player's hand and the dealer's upcard—the house edge in standard blackjack is approximately 0.5%, making it one of the lowest-edge casino games.^[31] This strategy minimizes losses over the long term by dictating when to hit, stand, double, or split, reducing the casino's advantage from around 2% for novice play to near 0.5%.^[31] Common variants alter these rules to vary the house edge and gameplay. In European blackjack, the dealer receives no hole card and must stand on all 17s, but peeking for blackjack is limited, increasing the house edge slightly by about 0.11% compared to American rules.^[31] Spanish 21, played with 48-card decks omitting all 10s, offers bonuses for specific hands like a 21 made with five or more cards (paying 3:2) or suited 21s (paying 2:1), which can make it player-friendly despite the reduced deck, though the house edge remains around 0.5-0.8% with proper strategy.^[31] The game's historical roots trace to 18th-century France as vingt-et-un (twenty-one), first documented around 1760 as an upper-class pursuit possibly evolved from earlier games like thirty-one, where the flexible ace value added strategic depth.^[32] It spread to Britain and America in the 19th century, retaining the name vingt-et-un or simply 21, with early English references appearing in 1800.^[32] The name "blackjack" emerged in the early 20th-century United States, likely from casino promotions offering special payouts for an ace of spades paired with a black jack.^[32] The game gained widespread popularity in US casinos following Nevada's legalization of gambling on March 19, 1931, which established regulated house-banked play and attracted players with bonus payouts to boost interest in 21 amid the Great Depression.^[33]^[34] By the mid-20th century, standardized rules solidified its status as a casino staple.^[33]

Other games and sports

The Game of 21 is an impartial combinatorial subtraction game played by two participants who alternate turns removing 1, 2, or 3 objects (such as coins or matches) from an initial pile of 21; the player who takes the last object wins under normal play convention.^[35] A misère variant exists where the player forced to take the last object loses, altering the endgame dynamics for smaller piles.^[36] This game exemplifies misère impartial games analyzed in combinatorial game theory, where optimal play depends on position classification as winning or losing. The winning strategy in the standard version relies on the Sprague-Grundy theorem, assigning nimbers (Grundy numbers) to positions; for the subtraction set {1,2,3}, these nimbers are periodic with period 4, making multiples of 4 (including 0) losing positions for the player about to move if the opponent plays perfectly.^[35] The first player secures victory by subtracting 1 on the opening turn to leave 20, then always responding to the opponent's move by subtracting the amount needed to reach the next multiple of 4 (e.g., if the opponent takes 2, subtract 2 to leave 16).^[35] In the misère form, the strategy shifts for piles of 1 or fewer than 4, but the first player still wins from 21 with optimal play by mirroring the normal strategy until the endgame adjustment.^[36] In professional sports, jersey number 21 has been iconically worn by athletes across disciplines, symbolizing excellence and longevity. Basketball power forward Tim Duncan donned 21 for all 19 seasons with the San Antonio Spurs from 1997 to 2016, earning five NBA championships, three Finals MVP awards, and two regular-season MVP honors while anchoring the franchise's dynasty.^[37] Similarly, in baseball, outfielder Roberto Clemente wore 21 throughout his 18-year career with the Pittsburgh Pirates from 1955 to 1972, amassing exactly 3,000 hits, 12 Gold Glove Awards, and the 1971 World Series MVP before his tragic death in a humanitarian mission.^[37] The number's significance extends to other sports, such as American football where running back LaDainian Tomlinson wore it while rushing for over 13,600 yards and scoring 145 rushing touchdowns. Table tennis formerly used a 21-point scoring system per game under International Table Tennis Federation (ITTF) rules from 1926 until 2001, requiring a two-point margin after 20-20 deuce; this was shortened to 11 points to accelerate rallies and enhance spectator appeal.^[38] Military-themed video games often simulate the 21-gun salute, a ceremonial tradition firing 21 rounds from cannons or rifles to honor dignitaries, fallen soldiers, or national events, rooted in naval customs dating to the 17th century. In Call of Duty: Advanced Warfare (2014), players encounter a scripted 21-gun salute during a comrade's funeral sequence, underscoring themes of loss and duty in the first-person shooter genre.^[39] Similar depictions appear in tactical simulators like Arma 3 (2013), where players can orchestrate the salute using in-game artillery for immersive role-playing.^[40]

Arts and entertainment

Film and television

In film and television, the number 21 has been prominently featured in titles and narratives, often symbolizing chance, loss, or transition, particularly in stories involving gambling, mortality, and youth. Notable works include heist dramas inspired by real-life exploits and introspective thrillers exploring human fragility. The 2008 film 21, directed by Robert Luketic, dramatizes the true story of the MIT Blackjack Team, a group of students who used card-counting techniques to win millions from Las Vegas casinos in the 1990s.^[41] The movie stars Jim Sturgess as Ben Campbell, a brilliant MIT student recruited by professor Micky Rosa (Kevin Spacey) to join the team, alongside Kate Bosworth, Liza Lapira, and Laurence Fishburne.^[42] Loosely based on the nonfiction book Bringing Down the House by Ben Mezrich, it highlights the high-stakes world of blackjack and the personal risks involved.^[43] The film grossed $159.8 million worldwide against a $35 million budget, becoming a commercial success and popularizing the MIT team's legend.^[41] 21 Grams (2003), directed by Alejandro González Iñárritu, is a nonlinear drama intertwining the lives of three characters devastated by a tragic car accident: mathematician Paul Rivers (Sean Penn), grieving widow Cristina Peck (Naomi Watts), and ex-convict Jack Jordan (Benicio del Toro).^[44] The film's title derives from the 1907 experiment by physician Duncan MacDougall, who claimed the human soul weighs 21 grams based on weight loss observed in dying patients—a notion now widely regarded as a debunked myth.^[45] Written by Guillermo Arriaga as part of Iñárritu's "Trilogy of Death," it earned critical acclaim for its emotional depth and performances, with Watts and del Toro receiving Academy Award nominations.^[46] The 21 Jump Street franchise originated as a television series (1987–1991), created by Patrick Hasburgh and Stephen J. Cannell, which followed a squad of young-looking undercover police officers posing as high school students to infiltrate teen-related crimes. Starring Johnny Depp as Officer Tom Hanson, the show addressed social issues like drugs and bullying across four seasons on Fox. It spawned a 2012 comedic film reboot directed by Phil Lord and Christopher Miller, starring Jonah Hill as Schmidt and Channing Tatum as Jenko—two bumbling cops reassigned to the Jump Street program to bust a high school drug ring.^[47] The movie parodies the original while updating its premise for humor, grossing over $201 million worldwide and leading to a 2014 sequel, 22 Jump Street. Beyond dedicated titles, 21 frequently appears as a plot device in television heist narratives, evoking blackjack's tension.

Music

Adele's second studio album, 21, released in January 2011, became one of the most commercially successful albums in modern history, selling over 31 million copies worldwide and establishing it as the best-selling album of the 21st century.^[48] The album features prominent tracks such as "Rolling in the Deep," which exemplifies its blend of soulful pop and heartbreak themes, contributing to its widespread acclaim.^[49] At the 54th Annual Grammy Awards in 2012, 21 won Album of the Year and Best Pop Vocal Album, while Adele swept six categories overall, marking a pinnacle of her career.^[49] In the UK, 21 held the number-one position for 23 weeks, the longest run for any album by a female solo artist at the time.^[50] In rock music, Green Day's "21 Guns," released in May 2009 as the second single from their album 21st Century Breakdown, addresses themes of surrender and inner conflict through its anthemic structure and lyrics.^[51] Similarly, Bad Religion's "21st Century (Digital Boy)," originally released on November 23, 1990, from their album Against the Grain, satirizes modern alienation and technological detachment in punk rock style; it was later re-recorded for Stranger Than Fiction in 1994.^[52] In classical music, Johannes Brahms composed the Hungarian Dances, a set of 21 lively pieces originally for piano four hands, published between 1869 and 1880 and inspired by Hungarian folk themes, with several later orchestrated.^[53] These works highlight the number 21 as a structural element in musical collections, distinct from direct titling.

Literature

In Patrick O'Brian's posthumously published novel 21: The Final Unfinished Voyage of Jack Aubrey (2004), the number 21 signifies the twenty-first installment in the Aubrey–Maturin series, chronicling the naval adventures of Captain Jack Aubrey and Dr. Stephen Maturin during the Napoleonic Wars; the manuscript, left incomplete at O'Brian's death in 2000, was edited and released by W. W. Norton & Company. William Pène du Bois's children's novel The Twenty-One Balloons (1947), a Newbery Honor book published by Viking Press, centers on Professor William Waterman Sherman, who embarks on a balloon voyage across the Pacific intended to last one year but crash-lands on the volcanic island of Krakatoa, where he encounters a utopian society powered by diamond mines and escapes using twenty massive hot-air balloons. In poetry, George Oppen's early collection 21 Poems (1923), discovered among Ezra Pound's papers and republished by New Directions in 2017, comprises 21 experimental verses by the 21-year-old Objectivist poet, exploring themes of urban modernity and personal fragmentation in pre-Depression America.^[54] Dante Alighieri's Inferno, the first part of The Divine Comedy (completed c. 1320), features Canto 21 as a pivotal episode in the Eighth Circle of Hell, where the pilgrim Dante observes the barrators—corrupt officials—submerged in boiling pitch and tormented by demons led by Malacoda, symbolizing the sticky entanglements of graft.^[55] The number 21 also recurs in speculative fiction, particularly in anthologies of 21st-century science fiction; for instance, Twenty-First Century Science Fiction (2013), edited by David G. Hartwell and Patrick Nielsen Hayden for Tor Books, gathers over 50 stories by authors like Elizabeth Bear and Cory Doctorow.^[56]

Culture and society

Symbolism and religion

In numerology, the number 21 is often interpreted as the product of 3 and 7, where 3 symbolizes creativity, expression, and growth, while 7 represents introspection, spirituality, and inner wisdom; this combination is seen to denote maturity, the completion of a cycle, and personal transformation.^[57] It is associated with advancement, success, and the realization of potential after challenges, reflecting a harmonious blend of outward achievement and inner development.^[58] In religious contexts, 21 holds significance in Tarot as the number of The World card in the Major Arcana, the final card symbolizing fulfillment, wholeness, integration, and the successful conclusion of a spiritual journey.^[59] The card depicts a dancing figure within a wreath, encircled by four elemental symbols, representing cosmic harmony and the attainment of enlightenment or unity with the universe.^[60] In Hindu traditions, 21 is linked to the 21 forms of the goddess Tara, a manifestation of the divine feminine energy known for protection, compassion, and liberation from suffering; these forms are invoked in rituals and tantric practices to address specific obstacles, with each Tara embodying unique attributes such as healing, wisdom, or wrathful power.^[61] In modern spiritual practices, particularly angel number interpretations, 21 is viewed as a message of positive change, alignment with one's higher purpose, and encouragement to maintain optimistic thoughts, as it combines the energies of new beginnings (1) and balance or partnerships (2).^[62] It signals that challenges are resolving, leading to abundance and personal growth when one trusts intuitive guidance.^[63] The age of 21 serves as the age of majority in several countries, including Bahrain, Cameroon, Chad, Egypt, Eswatini, Honduras, Lesotho, Madagascar, Niger, UAE, and others, totaling around 13 jurisdictions where individuals gain full legal adulthood at this age.^[64] In the United States, while the general age of majority is 18 in most states, certain rights such as purchasing handguns from licensed dealers are restricted to those 21 and older under the Gun Control Act of 1968. Similarly, Singapore maintains 21 as the age of majority for many purposes, including voting eligibility, despite lowering contractual capacity to 18 in 2009.^[65] The age of 21 is also a key threshold for alcohol consumption and purchase in numerous nations, with the United States enforcing a national minimum drinking age of 21 since the National Minimum Drinking Age Act of 1984, which conditioned federal highway funding on states adopting this standard—all states complied by 1988. This aligns with policies in about 17 countries, such as Indonesia, Sri Lanka, and Kazakhstan, where 21 marks the legal limit for alcohol acquisition. In the U.S., additional restrictions tied to 21 include the minimum age for casino gambling in most states, where tribal and commercial casinos typically bar entry and play to those under 21.^[66] Tobacco and nicotine product purchases are federally prohibited for those under 21 following the 2019 Tobacco 21 legislation, though enforcement varies by state. Celebrations of turning 21 hold cultural significance in Western societies as a rite of passage into full adulthood, often marked by parties, gifts, and symbolic gestures reflecting newfound independence. In the United Kingdom, a longstanding tradition involves presenting a "key to the door," symbolizing the freedom to come and go without parental oversight, a custom rooted in historical notions of maturity at 21.^[67] This milestone echoes broader Western customs, such as elaborate 21st birthday bashes in the U.S. and Australia, emphasizing legal permissions like drinking.^[68] Globally, 21 remains the voting age in nine countries, including Singapore and Bahrain, where citizens must reach this age to participate in national elections.^[69]^[70] In the U.S., some states impose driving-related rules tied to 21, such as requiring a supervising adult driver to be at least 21 years old during learner's permit practice for minors. The ratification of the 21st Amendment in 1933, which repealed Prohibition, further cemented 21's role in U.S. legal history by restoring alcohol regulation to the states without specifying an age, paving the way for later age-based restrictions.

Historical events

In 21 BC, Marcus Vipsanius Agrippa, a prominent Roman general and close ally of Emperor Augustus, married Julia the Elder, Augustus's daughter, in a politically motivated union aimed at strengthening the imperial succession line.^[71] This marriage produced several children, including Gaius and Lucius Caesar, who were later adopted by Augustus as potential heirs.^[71] The year 19 AD marked the assassination of Arminius, the Cheruscan chieftain who had led the Germanic tribes to victory over Roman forces in the Battle of the Teutoburg Forest in 9 AD. Killed by members of his own tribe who feared his rising power, Arminius's death ended a period of intense Roman-Germanic conflict along the Rhine frontier.^[72] The 21-gun salute emerged as a military and naval tradition in the 17th century, when European warships would fire volleys to signal peaceful intentions by depleting their ammunition stores, rendering themselves temporarily defenseless.^[73] Over time, the number of guns was standardized to 21 for salutes to sovereigns and heads of state, reflecting international agreements on honors; the United States Navy formalized this practice in 1875 through a treaty with Great Britain, establishing it as the highest tribute for visiting dignitaries.^[74] During World War I, Japan issued the Twenty-One Demands to the Republic of China on January 18, 1915, under Prime Minister Ōkuma Shigenobu, seeking exclusive economic privileges in Manchuria, control over key railways and ports, and veto power over China's foreign policy decisions.^[75] Presented amid Japan's seizure of German concessions in China, these demands provoked widespread protests in China, known as the May Fourth Movement, and escalated imperial rivalries that foreshadowed greater conflicts in Asia leading to World War II.^[76] The ratification of the 21st Amendment to the United States Constitution on December 5, 1933, repealed the 18th Amendment and brought an end to national Prohibition, which had banned the manufacture, sale, and transportation of alcohol since 1920.^[77] This amendment devolved regulatory authority over alcohol to the states, influencing ongoing legal frameworks for vice and commerce.^[78] The 21st century officially began on January 1, 2001, following the calendar convention that centuries start with the year ending in 01, rather than 00.^[79] This transition marked the start of a new era defined by rapid technological advancements, globalization, and geopolitical shifts, including the events of September 11, 2001.^[80]

References

[1]
21 - OeisWiki
Jul 20, 2017 · 21 is an integer, the smallest number of distinct squares needed to tile a square (see A006983). Contents. 1 Membership in core sequences; 2 ...
[2]
A000045 - OEIS
Fibonacci numbers are those numbers m such that m*phi is closer to an integer than k*phi for all k, 1 <= k < m. More formally: a(0)=0, a(1)=1, a(2)=1, a(n+1) = ...Missing: eighth | Show results with:eighth
[3]
Harshad Number -- from Wolfram MathWorld
... 21, 24, ... (OEIS A005349). Grundman (1994) proved that there is no sequence of more than 20 consecutive Harshad numbers, and found the smallest sequence of ...
[4]
A006983 - OEIS
Number of simple perfect squared squares of order n up to symmetry. (Formerly M4482). 29. 0, 0, 0, 0, 0, 0, 0, 0, ...
[5]
[PDF] Gun Salute - VA.gov
These cultures believed the number seven to be sacred and, therefore, it is believed, multiples of seven would be looked upon favorably (hence 21).
[6]
The 26th Amendment | Richard Nixon Museum and Library
Jun 17, 2021 · Ratified in July 1971, the 26th Amendment to the United States Constitution lowered the voting age of US citizens from 21 to 18 years old.
[7]
[PDF] Adulthood in Law and Culture - Scholarship Repository
The U.S. age of majority was lowered from twenty-one to eighteen, however, for reasons quite unrelated to capacity.
[8]
Blackjack
Blackjack (also known as twenty-one or sometimes pontoon) is one of the most popular casino card games in the world.
[9]
Strictly by the numbers movie review (2008) - Roger Ebert
Rating 1.5/4 · Review by Jim EmersonMar 27, 2008 · Ben Campbell (Jim Sturgess) attempts to beat the house in “21,” which was inspired by a real-life gambling scheme.
[10]
Semiprime -- from Wolfram MathWorld
The first few are 4, 6, 9, 10, 14, 15, 21, 22, ... (OEIS A001358). The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes) are 6, ...
[11]
Base Converter | Number conversion - RapidTables.com
Convert number from any base to any base. Enter number From Base 2 (binary) 3 4 5 6 7 8 (octal) 9 10 (decimal) 11 12 13 14 15 16 (hex)
[12]
Blum Integer - GeeksforGeeks
Jun 23, 2022 · Blum Integer is a semi-prime number, suppose p and q are the two factors (ie n = p * q), they(p and q) are of the form 4t + 3, where t is some integer.Missing: smallest | Show results with:smallest
[13]
A000217 - OEIS
Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n. (Formerly M2535 N1002). 4851. 0, 1, 3, 6, 10, 15, ...
[14]
A000567 - OEIS
A000567 - OEIS. (Greetings from The On-Line Encyclopedia of Integer Sequences!) Octagonal numbers: n*(3*n-2). Also called star numbers.
[15]
A001006 - OEIS
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021. Jackson ...
[16]
A000931 - OEIS
Padovan sequence (or Padovan numbers): a(n) = a(n-2) + a(n-3) with a(0) = 1 ... See Section 2, where the Padovan sequence is represented as a spiral of cubes (see ...
[17]
Perfect Square Dissection -- from Wolfram MathWorld
Skinner (1993) gives the smallest possible side length (and smallest order for each) as 110 (22), 112 (21), 120 (24), 139 (22), 140 (23), ... for simple perfect ...
[18]
Scandium | Sc (Element) - PubChem - NIH
It was discovered by the Swedish chemist Lars-Fredrik Nilson in 1879 in an ytterbium sample. In the same year, the Swedish chemist Per Theodore Cleve proved ...<|control11|><|separator|>
[19]
WebElements Periodic Table » Scandium » properties of free atoms
Scandium atoms have 21 electrons and the shell structure is 2.8.9.2. The ground state electron configuration of ground state gaseous neutral scandium is [Ar].
[20]
Scandium | Periodic Table | Thermo Fisher Scientific - US
Scandium • Transition Metal ; Melting point: 1814 K ; Boiling point: 3103 K ; Density[kg/m3]: 2985 ; Molar volume: 15.00 × 10-6 m3/mol ; Protons/Electrons: 21
[21]
Production of scandium radionuclides for theranostic applications
May 25, 2021 · Scandium isotopes with medical potential. Natural scandium has only one stable isotope, scandium-45 (45Sc). Several other scandium isotopes ...
[22]
Scandium - Element information, properties and uses | Periodic Table
It is the main component of the very rare and collectable mineral thortveitite, found in Scandinavia. Scandium can be recovered from thortveitite or ...
[23]
[PDF] The Properties and Application of Scandium-Reinforced Aluminum
The bicycle frames showed a 12% reduction in weight, a. 50% increase in yield strength, and a 24% improvement in fatigue life over the best-selling aluminum ...<|control11|><|separator|>
[24]
The Hydrogen 21-cm Line - HyperPhysics Concepts
The 21-cm wavelength line of hydrogen gas. At 1420 MHz, this radiation from hydrogen penetrates the dust clouds and gives us a more complete map of the ...
[25]
Exploring the cosmic dawn and epoch of reionization with 21cm line
Mar 14, 2023 · This epoch is called cosmic dawn. Cosmic reionization is the major transition of the intergalactic medium (IGM) in the universe driven by ...
[26]
Blackjack - Bicycle Cards
Blackjack's goal is to get as close to 21 as possible without going over. Players are dealt two cards, and the dealer one up and one down. Players can hit or ...
[27]
Blackjack - Wizard of Odds
### Standard Blackjack Rules
[28]
Blackjack: David Parlett's history of
Vingt-et-un (Twenty-One) first appears as an upper class or at least socially respectable game in 18th-century France, perhaps around 1760. It is not mentioned ...
[29]
History of Blackjack - The World of Playing Cards
Feb 11, 2010 · Most agree it probably originated in French casinos around 1700, where it was called Vingt-et-Un which, translated, means twenty-one.
[30]
The History of Blackjack and Card Counting
Blackjack came from games in France and Spain. The version that eventually swept across Europe was called “Vingt-un” or “Vingt-et-un” (meaning “21”).
[31]
[PDF] Theory of Impartial Games - MIT
Feb 3, 2009 · Let's look at another example game, called the 21 subtraction game. You and a friend start with 21 coins. You take turns taking up to 3 ...
[32]
[PDF] GAME THEORY - CMU School of Computer Science
Here are the rules of a very simple impartial combinatorial game of removing chips from a pile of chips. (1) There are two players. We label them I and II. (2) ...
[33]
From 00 to 99: Picking the top pro athletes to wear each uniform ...
Apr 17, 2017 · ... NFL; Rob Niedermayer, NHL; Josh Gibson, baseball. 21. Roberto Clemente, MLB; Tim Duncan, NBA; Warren Spahn, MLB; LaDainian Tomlinson, NFL ...
[34]
21s: An Adaptable Cricket Game for Skill Development
Dec 6, 2020 · Twenty ones is a simple, effective and scalable net/middle practice game designed to get players competing, thinking and improving.
[35]
HistoryofTableTennis - International Table Tennis Federation
2001, Game score changed from 21 to 11 points World Championships held in Osaka, Japan. Total triumph for China for the third time, winning all of the seven ...
[36]
Call of Duty: Press X to feel something - Polygon
Nov 4, 2014 · In the background a group of Marines were firing off a 21-gun salute. Apparently my character left the funeral of his best friend early. There ...
[37]
Arma 3 - 21 Gun Salute - YouTube
Dec 2, 2020 · Like my content and want to support me more directly? I'm on Patreon: http://www.patreon.com/dslyecxi ▻ About ShackTac: ...
[38]
21 (2008) - Box Office Mojo
Opening, Gross. Brazil, Apr 18, 2008, $621,090, $3,457,358. Mexico, Jun 20, 2008, $549,319, $3,059,180. Venezuela, Apr 25, 2008, $137,318, $796,011. Argentina ...
[39]
21 (2008) - IMDb
Rating 6.8/10 (277,633) 21 is about six MIT students who become trained to be experts in card counting in blackjack and subsequently took Vegas casinos for millions in winnings.Full cast & crew · Parents guide · Robert Luketic · Jim Sturgess as Ben
[40]
21 (2008) - Box Office and Financial Information - The Numbers
Opening Weekend: $24,105,943 (29.7% of total gross). Legs: 3.37 (domestic box office/biggest weekend). Domestic Share: 50.8% (domestic box office/worldwide).
[41]
21 Grams (2003) - IMDb
Rating 7.6/10 (251,599) 21 Grams tells the story of three lives brought together by tragedy: the life of a very sick man (Penn), a mother who has lost much (Watts), and a Christian ...Full cast & crew · Plot · Sean Penn as Paul Rivers · Filming & production
[42]
Was the Weight of a Human Soul Determined to Be 21 Grams?
Oct 26, 2003 · The claim that the soul weighs 21 grams is false. Experiments were flawed and inconsistent, and no credence should be given to the idea.Missing: myth | Show results with:myth
[43]
21 Grams - Rotten Tomatoes
Rating 80% (177) Paul Rivers (Sean Penn) is an academic dealing with a terminal heart condition, but his life is changed by a car crash that seems unrelated to his ailment. The ...
[44]
21 Jump Street | Rotten Tomatoes
Rating 85% (224) When cops Schmidt (Jonah Hill) and Jenko (Channing Tatum) join the secret Jump Street unit, they use their youthful appearances to go under cover as high- ...22 Jump Street · 224 Reviews · Cast and Crew · Trailers & Videos
[45]
Adele albums and songs sales - ChartMasters
Adele surpassed 119.9m equivalent album sales. · Her biggest album is 21, with 56.0m units, of which 30.4m in pure sales. · Her signature song is Someone Like You ...Missing: 2023 | Show results with:2023
[46]
For The Record: Adele's Icon-Making '21' At 10 - GRAMMY.com
To date, it has sold 7.2 million copies in America alone. It might not be the romantic tale Swift dreamed of growing up, but her sophomore album signalled that ...
[47]
21 – ADELE | Official Charts
21 opened with a 11-week stay as the UK's Official Number 1 album, and would go on to accumulate 23 weeks as chart-topper – the longest ever for a female solo ...
[48]
Green Day – 21 Guns Lyrics - Genius
“21 Guns”, the second single off 21st Century Breakdown, is a power ballad that questions war and shows a narrator looking for peace.What is the meaning of "21... · Who wrote “21 Guns” by Green...
[49]
Bad Religion – 21st Century (Digital Boy) Lyrics - Genius
Greg Graffin. Released on. November 23, 1990. Songs That Sample 21st Century (Digital Boy). 1990 by The Hood Internet. 21st Century (Digital Boy) Interpolations.
[50]
Hungarian Dances | Romantic Era, Orchestral, Piano Duets
Oct 18, 2025 · Hungarian Dances, set of 21 dances composed by Johannes Brahms. Originally intended for two pianists, the dances were published in that form in two sets in ...
[51]
21 Poems by George Oppen - New Directions Publishing
Aug 15, 2017 · Four years earlier, the twenty-one-year-old poet had sent an unbound sheaf of typewritten poems with the title 21 Poems handwritten in pencil on
[52]
Inferno 21 - Digital Dante - Columbia University
They are truculent and defiant. They sing in only one key: that of resistance and opposition. Effectively, what they communicate is: “no, you may not pass, we ...
[53]
https://www.britannica.com/topic/Hungarian-Dances
[54]
Number 21 Meaning - Affinity Numerology
The numerology number 21 is both a creative spirit and a mostly reliable partner. In situations where the two conflict, the creative spirit generally prevails.
[55]
A4 - Compound Numbers • Practical Numerology
21. This number represents advancement, success, elevation and honor in life. It also stands for victory after a long fight. 22. This ...<|separator|>
[56]
https://us.macmillan.com/books/9780765326010/twentyfirstcenturysciencefiction
[57]
The World Tarot Card Meanings
This card could represent graduation, a marriage, the birth of a child, or achieving a long-held dream or aspiration. You have finally accomplished your goal or ...
[58]
https://www.astronumero.org/compound-numbers/
[59]
Two thousand - 21 - Trattoria Zà Zà
Dec 29, 2020 · 21 is the number of fidelity, trust and loyalty and also of courage, true friendships, laws and civil rights.Brave actions are connected with number 21.
[60]
Angel Number 21 Meaning - A Symbol of Independence
The Angel Number 21 symbol forces you to convince yourself and assert positively about your determination to succeed in your ambitions.
[61]
https://www.exoticindiaart.com/article/21-forms-of-goddess-tara/
[62]
Age of Majority by Country 2025 - World Population Review
In 2025, the age of majority varies. Some countries like Madagascar, Ivory Coast, and Cameroon are 21, while many others, including the US, China, and Japan, ...
[63]
Civil Law Act 1909 - Singapore Statutes Online
As from 1 March 2009, a contract entered into by a minor who has attained the age of 18 years shall have effect as if he were of full age.
[64]
Voting rights in Singapore - Wikipedia
All Singapore citizens not less than 21 years old on the cut-off date for the registration of electors (1 January of a particular year), and ordinarily resident ...<|control11|><|separator|>
[65]
https://sso.agc.gov.sg/Act/CLA1909?ProvIds=pr35-
[66]
Legal drinking age - Wikipedia
In Mexico, the drinking age is 18 in all states. · In the United States, the minimum legal age to purchase alcoholic beverages is mainly 21 years of age; the two ...
[67]
Gambling Age by State 2025 - World Population Review
California's minimum age for casinos and poker rooms is 18-21 depending upon the establishment. · Georgia's minimum age of 18 for casinos applies only to cruises ...
[68]
21st Birthday Key Of The Door - Historic UK
Feb 4, 2021 · The 'key to the door' signified the reaching of adulthood and the independence that brought with it. Of course it is not so significant these days.
[69]
Birthday Celebration Traditions in the UK - Cakes Overseas Blog
Oct 31, 2022 · The 21st is considered the legitimate 'coming of age', and it comes with more elegant celebrations such as a dress-up dinner with friends and ...
[70]
Voting Age by Country 2025 - World Population Review
Some municipalities allow 16-year-olds to vote in local elections. Pakistan Flag. Pakistan, 18. Reserved parliamentary seats for women ...
[71]
Bahrain - ACE Electoral Knowledge Network
Question: What is the legal voting age in the national elections? Answer: f. 21. Comments: Citizens should be 21 years old on the day of referendum or ...
[72]
Julia | Roman Empress, Adopted Heir, Political Strategist | Britannica
Oct 27, 2025 · After a brief marriage to Marcus Marcellus, who died in 23 bc, Julia wedded Marcus Vipsanius Agrippa, Augustus' chief lieutenant, in 21 bc.
[73]
Twenty-One Gun Salute - Naval History and Heritage Command
By 1730, the Royal Navy was prescribing twenty-one guns for certain anniversary dates, although this was not mandatory as a salute to the royal family until ...
[74]
21 Gun Salute - Marine Corps University
The 21-gun salute, a British custom, is the highest national honor, rendered to the US President, ex-Presidents, and foreign heads of state. It is also given ...
[75]
Historical Documents - Office of the Historian
Relations between China and Japan; “Twenty-one demands” on China by Japan; attitude of the United States. Extradition of John Eills for perjury refused by ...
[76]
China, Japan, and the Twenty-One Demands
Jan 21, 2015 · Japan presented Chinese President Yuan Shikai (Yuan Shih-Kai) with requests that would have turned China into a de facto Japanese protectorate.
[77]
21st Amendment - Repeal of Prohibition | Constitution Center
Section 1. The eighteenth article of amendment to the Constitution of the United States is hereby repealed.
[78]
The Repeal of Prohibition
... 21st Amendment that would repeal the 18th. The House and Senate passed it in February 1933 and sent it to the states for final approval. Under the amendment ...
[79]
The 21st Century and the 3rd Millennium
The 21st century began on 1 January 2001 and will continue through 31 December 2100. Similarly, the 1st millennium comprised the years AD 1-1000.
[80]
Did the Millennium Start in Year 2000 or 2001? - Time and Date
Jan 1, 2000 · The 21st Century Started in 2001. In 1999, the world was preparing for the New Year's party of a lifetime. The year number in the Gregorian ...

21