64 (number)

Sixty-four (64) is the natural number following 63 and preceding 65. It is the sixth power of two (2⁶ = 64), the square of eight (8² = 64), and the cube of four (4³ = 64).^[1][2] As an even composite number, 64 has the prime factorization 2⁶ and exactly seven positive divisors: 1, 2, 4, 8, 16, 32, and 64.^[3] In mathematics and computing, 64 holds particular significance as a power of two, enabling efficient binary representations and memory addressing. It defines 64-bit architectures, where processors handle 64-bit data widths, allowing access to vastly larger memory spaces compared to 32-bit systems; the MIPS R4000, introduced in 1991, was the first commercial 64-bit microprocessor.^[4] The Commodore 64, released in 1982, became the best-selling single computer model ever, with between 12.5 and 17 million units sold, thanks to its low cost, advanced graphics, and extensive software library.^[5] Similarly, the Nintendo 64 console, launched in 1996, pioneered 3D gaming with titles like Super Mario 64 and sold 32.93 million units worldwide.^[6][7] Beyond technology, 64 appears in science and culture. In chemistry, it is the atomic number of gadolinium (Gd), a lanthanide metal used in nuclear reactors for neutron absorption and in MRI contrast agents due to its magnetic properties.^[8] Biology recognizes 64 as the number of possible codons in the genetic code, formed by triplets of the four nucleotide bases (A, C, G, U/T), which specify 20 amino acids and stop signals across all life forms.^[9] Culturally, a standard chessboard consists of an 8×8 grid totaling 64 alternating light and dark squares, foundational to the game's strategy since its origins.^[10] In ancient Chinese philosophy, the I Ching (Book of Changes) comprises 64 hexagrams, each a unique combination of six yin-yang lines derived from eight trigrams, symbolizing cosmic patterns and used for divination over millennia.^[11]

In mathematics

Basic properties

Sixty-four is the natural number following 63 and preceding 65. It is the cardinal number sixty-four and the ordinal number sixty-fourth.^[12][13] The prime factorization of 64 is $2^6$26.^[14] The positive divisors of 64 are 1, 2, 4, 8, 16, 32, and 64, totaling seven divisors.^[15] In various numeral systems, 64 is represented as $1000000_2$10000002​ in binary, $100_8$1008​ in octal, and $40_{16}$4016​ in hexadecimal.^[16] Additionally, 64 can be expressed as the square of 8 ($8^2 = 64$82=64), the cube of 4 ($4^3 = 64$43=64), and the sixth power of 2 ($2^6 = 64$26=64).^[14]

Number theory

64 is the seventeenth interprime number, positioned midway between the consecutive primes 61 and 67, as it equals their arithmetic mean.^[17] The aliquot sum of 64, which is the sum of its proper divisors excluding itself, is calculated as $1 + 2 + 4 + 8 + 16 + 32 = 63$1+2+4+8+16+32=63. This value being less than 64 classifies it as a deficient number.^[18] In the context of additive representations, 64 is the seventh Erdős–Woods number, meaning it serves as the length of an interval of 64 consecutive integers where each integer shares a common factor with at least one of the endpoints of the interval.^[19] 64 is also the tenth self number in base 10, a positive integer that cannot be expressed as $n$n plus the sum of the digits of $n$n for any positive integer $n$n. The first ten self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, and 64.^[20] As a power of 2, specifically $2^6$26, 64 appears in the sequence of powers of 2 cataloged as OEIS A000079, where it is the term for $n=6$n=6.^[21]

Geometry and combinatorics

In geometry, the number 64 prominently appears in the structure of the standard chessboard, which consists of an 8×8 grid of alternating black and white squares, totaling 64 unit squares that form the playing field for the game of chess.^[22] This grid exemplifies a planar tessellation by squares, where the even division into 64 equal parts facilitates symmetric movement rules for pieces like the rook along rows and columns. Higher-dimensional geometry further highlights 64's significance. The 6-dimensional hypercube, or hexeract, possesses 64 vertices, each corresponding to a unique combination of coordinates with entries of 0 or 1 in six dimensions, underscoring 64 as $2^6$26.^[23] This structure extends the tesseract (4D hypercube with 16 vertices) and illustrates how powers of 2 enumerate vertices in n-dimensional cubes. In four-dimensional space, there are 64 convex uniform polychora—vertex-transitive polytopes with regular polyhedral cells—excluding infinite prismatic families; these include the six regular ones like the 5-cell and 120-cell, cataloged through systematic enumeration of Wythoff constructions and symmetry groups.^[24] Additionally, four-dimensional crystallography recognizes 64 Bravais lattices, which classify periodic point arrangements up to translation, encompassing primitive and centered types derived from the 10 hypersymmetry systems.^[25] Combinatorially, 64 arises in enumerative problems related to geometric configurations, such as the 64 vertices of the chessboard graph, where squares are nodes and edges connect adjacent positions (e.g., for rook or king moves), enabling counts of paths, colorings, and matchings that model strategic placements.^[26] In graph theory contexts, configurations like the knight's tour on an 8×8 board explore Hamiltonian paths across these 64 positions, with the total edges in the knight's graph summing to 336, reflecting the degree variations (2 to 8) across the board.^[27] These enumerations emphasize 64's role in counting discrete structures without delving into exhaustive proofs, prioritizing the scale of possibilities in bounded grids.

In science

Chemistry

Gadolinium (Gd) is the chemical element with atomic number 64 in the periodic table.^[28] It belongs to the lanthanide series of rare-earth metals and is positioned in the f-block, reflecting its characteristic inner transition metal properties.^[29] The electron configuration of gadolinium is $[\ce{Xe}] 4f^7 5d^1 6s^2$[\ceXe]4f75d16s2, which contributes to its unique magnetic and chemical behaviors.^[29] Gadolinium appears as a silvery-white metal with a bright metallic luster; it is both malleable and ductile, allowing it to be shaped without fracturing under mechanical stress.^[30] Below its Curie temperature of approximately 20 °C, gadolinium displays ferromagnetic properties, making it strongly attracted to magnetic fields—stronger than iron or nickel in this state.^[31] The element was first isolated in 1880 by Swiss chemist Jean Charles Galissard de Marignac, who identified its oxide in a sample of the mineral samarskite through spectroscopic analysis.^[29] Naturally occurring gadolinium comprises seven stable isotopes, with atomic masses ranging from 152 to 160: $^{152}$152Gd (0.20% abundance), $^{154}$154Gd (2.18%), $^{155}$155Gd (14.80%), $^{156}$156Gd (20.47%), $^{157}$157Gd (15.65%), $^{158}$158Gd (24.84%), and $^{160}$160Gd (21.86%).^[32] These isotopes, particularly $^{155}$155Gd and $^{157}$157Gd, exhibit high thermal neutron capture cross-sections, which underpin gadolinium's utility in nuclear applications. In chemical applications, gadolinium compounds serve as contrast agents in magnetic resonance imaging (MRI), enhancing image clarity through their paramagnetic effects without relying on radioactive isotopes.^[33] Gadolinium-based phosphors are integral to color television screens and X-ray scintillation detectors, where they convert energy into visible light for display and imaging purposes.^[34] Furthermore, gadolinium acts as a burnable neutron poison in commercial nuclear reactors, such as CANDU types, to regulate fission rates and maintain core stability during operation.^[35]

Biology

In biology, the number 64 is fundamentally linked to the structure of the genetic code, which governs the translation of genetic information into proteins. The standard genetic code consists of 64 possible codons, each a triplet of nucleotide bases that specify amino acids or termination signals during protein synthesis. These codons arise from the four canonical RNA bases—adenine (A), cytosine (C), guanine (G), and uracil (U)—arranged in three positions, yielding $4^3 = 64$43=64 unique combinations. The codon table organizes these 64 triplets such that 61 are sense codons, which encode one of the 20 standard amino acids, while the remaining three serve as stop codons (UAA, UAG, and UGA) to signal the end of translation. This redundancy, or degeneracy, means most amino acids are specified by multiple codons—ranging from one (for methionine and tryptophan) to six (for leucine, serine, and arginine)—allowing flexibility in the genetic sequence without altering the protein product.^[36] The elucidation of this 64-codon code occurred in the 1960s through pioneering experiments led by Marshall Nirenberg and Har Gobind Khorana. Nirenberg’s 1961 cell-free system demonstrated that the codon UUU codes for phenylalanine, marking the first decoding step, while Khorana’s synthesis of artificial RNA polymers enabled systematic assignment of all codons by 1966.^[37][38] The implications of this 64-codon framework are profound, underpinning the near-universality of the genetic code across bacteria, archaea, eukaryotes, and viruses, which suggests a common evolutionary origin predating the last universal common ancestor.^[39] This universality facilitates horizontal gene transfer and underscores the code's robustness. Additionally, the wobble hypothesis, proposed by Francis Crick in 1966, explains the observed degeneracy by allowing non-standard base pairing at the third codon position, enabling a reduced set of transfer RNAs (tRNAs) to recognize multiple codons efficiently.

Physics

In crystallography, the symmetry properties of three-dimensional crystals are classified into 32 point groups, which account for the possible combinations of rotation axes, mirror planes, and inversion centers compatible with lattice periodicity. These groups form the foundation for understanding crystal structures and their physical properties, such as optical birefringence and piezoelectricity. In higher dimensions, relevant to theoretical models of quasicrystals and multidimensional lattices, four-dimensional space features 64 Bravais lattice classes, comprising primitive and centered types that generalize the 14 three-dimensional Bravais lattices used in conventional solid-state physics. These structures provide a framework for exploring symmetry in hypothetical four-dimensional crystals, influencing studies of defects and phase transitions in advanced materials. The 64-cell, known as the hexeract, is the six-dimensional hypercube polytope with 64 bounding cubic facets, 192 square facets, and 64 vertices. In theoretical physics, such polytopes model compactified extra dimensions in unified theories, offering geometric insights into spacetime topology beyond four dimensions. Additionally, the Clifford algebra Cl(6) over the reals has dimension 64 and underpins the mathematical description of spinors in six-dimensional spacetime, essential for formulating fermionic fields in quantum field theory and extensions of the Dirac equation to higher dimensions. This algebra facilitates the representation of rotations and Lorentz transformations in models involving extra dimensions, such as those in supergravity.^[40] In particle physics, symmetry groups like SU(8) appear in grand unified theories and flavor models, where representations or tensor products yield 64-dimensional spaces, aiding the unification of quarks, leptons, and their interactions under a single gauge structure.^[41]

In computing and technology

Computing architectures

In computing, 64-bit architectures refer to processor designs where the fundamental unit of data, known as a word, is 64 bits wide, enabling operations on larger integers, floating-point numbers, and memory addresses compared to 32-bit systems, which are restricted to 4 gigabytes of directly addressable memory.^[42] This word size supports enhanced performance in data-intensive tasks through wider registers and buses.^[43] The origins of 64-bit computing trace back to supercomputers in the 1970s, with the Cray-1, introduced in 1976, featuring 64-bit integer and floating-point operations as a core element of its vector-processing design to handle complex scientific simulations.^[44] Widespread adoption occurred in the early 2000s for general-purpose computing, beginning with AMD's x86-64 (also called AMD64) architecture released in 2003 via the Opteron processor, which extended the 32-bit x86 instruction set while maintaining backward compatibility.^[45] Similarly, the ARM64 (AArch64) architecture emerged in 2011 with the ARMv8-A specification, optimizing 64-bit processing for power-efficient devices like smartphones and servers.^[46] A key advantage of 64-bit architectures is their expanded address space: processors can theoretically address up to $2^{64}$264 bytes, or 16 exabytes, vastly exceeding the needs of most applications and enabling robust handling of massive datasets. In contrast, 32-bit systems cap at $2^{32}$232 bytes (4 GB), often leading to memory fragmentation in demanding environments.^[42] These architectures underpin modern operating systems, including 64-bit variants of Windows and Linux, which leverage the full 64-bit address space for advanced memory management and virtual addressing. For instance, 64-bit Windows, first available in server editions like Windows Server 2003 and expanded to desktops with Windows Vista in 2007, supports configurations exceeding hundreds of gigabytes of RAM for enterprise workloads.^[47] Linux kernels have provided 64-bit support for various platforms, including Alpha since the 1990s and x86-64 since 2003, facilitating efficient virtual memory paging and process isolation in servers and desktops.^[48] The choice of 64 bits stems from its status as $2^6$26, a power of two that aligns seamlessly with binary representation and byte-oriented storage, where a standard byte is 8 bits ($2^3$23); thus, 64 bits equate to exactly 8 bytes, optimizing data alignment, caching, and arithmetic operations in hardware.^[49]

Video games and electronics

The Nintendo 64 (N64), released in 1996, was a pioneering home video game console equipped with a 64-bit NEC VR4300 CPU operating at 93.75 MHz, marking one of the first consumer gaming systems to employ 64-bit processing for enhanced 3D graphics capabilities.^[50] This architecture, derived from the MIPS R4300i, enabled smoother rendering and more complex game worlds compared to its 32-bit contemporaries. The console achieved global sales of 32.93 million units over its lifetime, driven by innovative titles such as Super Mario 64, which revolutionized 3D platforming by leveraging the system's hardware for free-roaming exploration and precise controls.^[7] Unlike competitors that adopted optical discs, the N64 utilized a cartridge-based storage system, which allowed for faster load times and built-in anti-piracy measures but limited game sizes to a maximum of around 64 MB due to manufacturing constraints.^[50] This design choice influenced game development toward optimized, texture-efficient 3D environments, as seen in landmark releases like The Legend of Zelda: Ocarina of Time. The N64's legacy extends to hardware innovations, particularly its three-pronged controller featuring the first widely adopted analog stick, which provided variable input sensitivity essential for navigating 3D spaces and inspired subsequent controller designs across the industry.^[51][52] In broader electronics, the number 64 appeared in memory standards during the late 1990s and early 2000s, when 64 MB of RAM emerged as a typical configuration for personal computers, supporting the transition to more demanding applications like early multimedia software and internet browsing.^[53] Hardware interfaces also commonly incorporated 64-pin connectors, such as those in specialized SIMM memory modules used in systems like certain Apple Macintosh models and Amiga accelerators, facilitating reliable data transfer in compact form factors.^[54] In graphics technology, 64-bit color depth represents a theoretical maximum of $2^{64}$264 (approximately 18.4 quintillion) colors, typically achieved through 48 bits for RGB channels (16 bits each) plus a 16-bit alpha channel for transparency, though practical implementations remain rare due to diminishing returns beyond 24-bit true color for human perception.^[55] This depth has been explored in high-end professional imaging and scientific visualization but sees limited use in consumer electronics, where bandwidth and storage demands outweigh visual benefits.

In culture and history

Religion and mythology

In ancient Chinese philosophy and divination, the number 64 holds profound significance through the I Ching (Book of Changes), an foundational text dating to the Western Zhou period (1046–771 BCE). The I Ching comprises 64 hexagrams, each formed by combining two of the eight trigrams—symbolic representations of natural forces and cosmic processes derived from yin-yang dualities. These hexagrams serve as oracular symbols for interpreting change, guiding ethical decisions, and understanding the universe's dynamic patterns, with each hexagram offering statements on situational meanings and transformative advice.^[11] In Hinduism, particularly within Tantric traditions, 64 manifests as the sacred count of Yoginis, female deities or enlightened practitioners embodying Shakti's (divine feminine energy) myriad aspects. The 64 Yoginis, often depicted in therianthropic forms with animal mounts symbolizing primal forces, emerged as manifestations of the goddess Devi (Durga) from her body during cosmic battles, representing esoteric powers tied to yoga, sorcery, and fertility rituals. Worshipped in circular open-air temples from the 9th to 14th centuries CE in regions like Odisha, they form mandala-like assemblies that invoke harmony between human, natural, and divine realms, emphasizing spiritual enlightenment through the awakening of kundalini energy at the muladhara chakra.^[56] Hindu texts further associate 64 with the chatuṣṣaṣṭi kalā (64 arts or skills), outlined in Vātsyāyana's Kāma Sūtra (c. 3rd–4th century CE) as essential accomplishments for cultivating aesthetic, intellectual, and sensual refinement. These arts encompass vocal and instrumental music, dance, painting, poetry recitation, carpentry, garland-making, and even omens interpretation, serving to enhance personal allure, social harmony, and the pursuit of kāma (pleasure) as one of life's four aims (puruṣārthas). Mastery of these kalās was deemed vital for courtesans, wives, and householders to foster relationships and self-sufficiency, mirroring the Tantric view of integrated worldly and spiritual practice.^[57] Tibetan Buddhism incorporates 64 in its Tantric frameworks, drawing from Indian Kaula traditions where 64 Tantras represent core esoteric scriptures on ritual, meditation, and deity yoga for realizing non-dual enlightenment. These Tantras, including influential texts like the Guhyasamāja and Cakrasaṃvara, emphasize the 64 Yoginis or dakinis as enlightened feminine energies in mandalas, such as the Kalachakra system's 64 speech goddesses emanating from the navel chakra to channel transformative wisdom and subtle body energies. This integration reflects the syncretic adoption of Hindu Tantric elements into Vajrayana practices by the 10th century CE, prioritizing secrecy and guru transmission for overcoming dualistic ignorance.^[58][59] Symbolically, 64 evokes completeness and cosmic totality across these traditions, as the product of 8×8—where 8 signifies renewal or infinity (e.g., the trigrams' permutations)—or the cube of 4, representing the elevation of the four classical elements (earth, water, fire, air) to a higher, perfected state in numerological interpretations. This squared structure underscores wholeness, from the I Ching's exhaustive mapping of change to Tantra's full spectrum of divine emanations.^[60]

Historical events

In 64 BC, the Roman general Pompey the Great concluded his campaigns in the eastern Mediterranean by annexing the remnants of the Seleucid Empire, formally incorporating Syria as a Roman province and extending Roman influence toward Judea.^[61] This expansion set the stage for further interventions in the region, including the resolution of internal Jewish conflicts between rival claimants to the Hasmonean throne. During this period, the Roman poet Gaius Valerius Catullus, born around 84 BC, was emerging as a significant literary figure in Verona and Rome, though no major personal milestone is recorded for that exact year.^[62] The year 64 AD is most notably marked by the Great Fire of Rome, a catastrophic blaze that began on July 18 and raged for six days, damaging much of the city including 10 of its 14 districts, with three completely destroyed.^[63] Emperor Nero, who was in Antium at the time, returned to organize relief efforts but faced accusations of arson to clear space for his Domus Aurea palace; in response, he scapegoated the Christian community, initiating the first widespread persecution of Christians in the empire through arrests, tortures, and executions.^[64] This event, described by ancient historians like Tacitus, exacerbated tensions between Roman authorities and the nascent Christian sect. The aftermath in 65 AD saw the forced suicides of Nero's former advisor Seneca the Younger and the poet Lucan (Marcus Annaeus Lucanus), both implicated in the Pisonian conspiracy against the emperor, reflecting the ongoing political instability following the fire.^[65][66] In 1864, amid the American Civil War, the Battle of Mobile Bay on August 5 represented a decisive Union naval victory, where Admiral David Farragut's fleet captured the Confederate stronghold at Mobile, Alabama, effectively closing the last major Confederate port on the Gulf of Mexico to blockade runners.^[67] Farragut's famous command, "Damn the torpedoes, full speed ahead," underscored the bold tactics that led to the surrender of the CSS Tennessee and the forts guarding the bay.^[68] This engagement, part of the broader Anaconda Plan to strangle the Confederacy's economy, contributed significantly to the Union's momentum heading into the war's final year. The year 1964 featured pivotal global developments, including the Gulf of Tonkin incident on August 2 and 4, when U.S. Navy destroyers reported attacks by North Vietnamese forces off Vietnam's coast, prompting President Lyndon B. Johnson to seek and obtain the Gulf of Tonkin Resolution from Congress on August 7, which authorized escalated U.S. military involvement in the Vietnam War.^[69] Culturally, the Beatles' arrival in the United States on February 7 and their debut performance on The Ed Sullivan Show on February 9 drew an estimated 73 million viewers, igniting Beatlemania and marking a transformative moment in global popular music. The number 64 has appeared in historical administrative contexts, such as in the United States, where Colorado is divided into 64 counties, a structure that evolved from its territorial days in 1861 with ongoing subdivisions to accommodate growth and governance needs.^[70] Similarly, Louisiana's 64 parishes—equivalent to counties in other states—trace their origins to early 19th-century French colonial influences and were formalized in the state's 1812 constitution, reflecting unique regional divisions post-Louisiana Purchase.^[71]

Arts and popular culture

In literature, the number 64 serves as a structural motif in several works, often symbolizing completeness or multiplicity. For instance, Will Buckingham's Sixty-Four Chance Pieces (2013) comprises sixty-four interconnected stories inspired by the hexagrams of the I Ching, exploring themes of chance and transformation through fable and travel narrative.^[72] Similarly, John Keats's The 64 Sonnets collects all sixty-four sonnets the poet composed between 1814 and 1819, highlighting his early romantic experimentation in verse form.^[73] In science fiction, the Sector 64 trilogy by Dean M. Cole (2014–2015) uses the number to denote a pivotal spatial sector in a thriller involving time travel and extraterrestrial threats, emphasizing navigational precision in higher-dimensional settings.^[74] In film and television, 64 frequently appears in contexts tied to strategy and confinement, particularly through chess imagery. The 2015 short film 64 Squares, directed by Steven S. Goers, depicts a tense chess match between a writer and a store employee, where the board's sixty-four squares become a metaphor for psychological confrontation and hidden identities.^[75] This theme recurs in broader media, as seen in the 2020 biographical drama Critical Thinking, which portrays the Miami Jackson High School chess team's rise and underscores the game's universality across sixty-four squares, regardless of socioeconomic background.^[76] References to Route 64 also emerge in episodic content, such as the PBS series North Carolina Weekend episode "Highway 64" (2021), which profiles roadside culture and communities along the historic U.S. Route 64, evoking themes of American wanderlust.^[77] The animated children's series 64 Zoo Lane (1999–2013), created by An Vrombaut, features a girl named Lucy who lives at 64 Zoo Lane adjacent to a zoo and shares bedtime stories with the anthropomorphic animal residents, incorporating the number 64 directly into its title and setting.^[78] Music often incorporates 64 through titles evoking aging or rhythm. The Beatles' "When I'm Sixty-Four" (1967), from the album Sgt. Pepper's Lonely Hearts Club Band, whimsically anticipates middle age and domestic life, becoming a cultural touchstone for generational reflection with its clarinet-driven vaudeville style.^[79] Additionally, the electronic artist 64 Beats produces tracks centered on steady, atmospheric rhythms, with releases like Lost Tape (2020) maintaining tempos around 64 beats per minute to evoke meditative pulses in ambient genres.^[80] In sports, 64 denotes scale in tournament formats and player identities. The NCAA Men's Basketball Tournament expanded to 64 teams in 1985 and to its current format of 68 teams in 2011 (as of 2025), transforming March Madness into a bracket spectacle that amplifies underdog narratives and national rivalries across multiple rounds.^[81] Jersey number 64 has been worn by Hall of Famers like guard Jerry Kramer of the Green Bay Packers (1958–1968), whose blocking prowess contributed to five NFL championships, and offensive lineman Randall McDaniel of the Minnesota Vikings (1988–1999), known for his durability in over 200 consecutive starts.^[82] Beyond these domains, 64 permeates modern pop culture through numerological interpretations and digital folklore. As an "angel number," 64 symbolizes stability, hard work, and divine support for material goals, often cited in self-help contexts as a sign to balance practicality with optimism during transitions.^[83] In memes and online humor, it frequently appears in 64th birthday cards poking fun at aging—such as chalkboard-style graphics joking about hexadecimal conversions or "not looking a day over 40 in hex"—while Nintendo 64 nostalgia fuels retro gaming memes celebrating the console's 1996 launch and iconic titles like Super Mario 64.^[84]