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82

Eighty-two is the natural number following 81 and preceding 83. It is an even composite number, the eighth term of the sequence of composite numbers (after 4, 6, 8, 9, 10, 12, 14, 15), and its prime factorization is 2 × 41. As a pronic number, it equals 9 × 9 + 1, but more notably, it is the product of the first two primes greater than 3 (5 and 7? Wait, no: actually 2×41). In base 10, 82 has various representations, including as a repunit in certain bases or in recreational math contexts. In number theory, 82 appears in properties related to totients and divisors, while in other mathematical contexts, it features in geometry and combinatorics. Scientifically, 82 is the atomic number of the chemical element lead (Pb).^[1] In astronomy, it relates to Messier object M82, a galaxy. Beyond math and science, 82 holds significance in sports (e.g., as a jersey number), transportation (e.g., Interstate 82), geography, and popular culture.

As a natural number

Properties

Eighty-two is an even composite natural number, divisible by two distinct primes.^[2] Its proper divisors are 1, 2, and 41, which sum to an aliquot sum of 44, classifying it as a deficient number since this sum is less than 82 itself.^[3] The sum of its digits in base 10 is 8 + 2 = 10.^[3] Iteratively summing these digits yields a digital root of 1.^[3] The number 82 is not a perfect square, as \sqrt{82} \approx 9.055.^[4] Likewise, it is not a perfect cube, with

\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{82} \approx 4.344

.^[5]

Representations

In positional numeral systems, the number 82 is expressed using digits that represent powers of the system's base. In the binary system (base 2), where only the digits 0 and 1 are used, 82 is represented as $1010010_2.^[6] This notation corresponds to $1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = [64](/page/64) + 16 + 2 = 82.^[6] In the octal system (base 8), using digits from 0 to 7, 82 is denoted as $122_8.^[7] This expands to $1 \times 8^2 + 2 \times 8^1 + 2 \times 8^0 = [64](/page/64) + [16](/page/16) + 2 = [82](/page/82).^[7] The hexadecimal system (base 16) employs digits 0-9 and A-F (where A=10, B=11, etc.), representing 82 as $52_{16}.^[8] Here, $5 \times 16^1 + 2 \times 16^0 = 80 + 2 = [82](/page/82).^[8] For illustration in a non-standard base, in the ternary system (base 3) with digits 0, 1, and 2, 82 is $10001_3.^[9] This equals $1 \times 3^4 + 0 \times 3^3 + 0 \times 3^2 + 0 \times 3^1 + 1 \times 3^0 = 81 + 1 = [82](/page/82).^[9] In the familiar decimal system (base 10), it is simply written as 82, using digits 0-9 to denote $8 \times 10^1 + 2 \times 10^0.

In mathematics

Number theory

In number theory, 82 has the prime factorization $82 = 2 \times 41, where both factors are prime numbers, making it a semiprime (a product of exactly two primes).^[10] It is the 27th semiprime in the ordered sequence of such numbers.^[11] The positive divisors of 82 are 1, 2, 41, and 82, giving it exactly four divisors. Euler's totient function evaluates to \phi([82](/page/82)) = 40, representing the count of positive integers up to 82 that are coprime to 82.^[12] As a semiprime of the form $2q where q = 41 is an odd prime, 82 is the 12th such number in ascending order (following 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, and 74). Its aliquot sequence, generated by iteratively applying the sum of proper divisors function, is 82 → 44 → 40 → 50 → 43, where 43 is prime and the sequence terminates thereafter (reaching 1 and then 0).^[13] This forms an aliquot sequence of length four before hitting a prime, consistent with its deficient nature (as the sum of proper divisors is less than 82).^[13]

Other contexts

82 is a happy number, a concept from recreational mathematics where a positive integer is considered happy if repeatedly replacing it with the sum of the squares of its digits eventually reaches 1. Starting with 82, the process proceeds as follows: $8^2 + 2^2 = 64 + 4 = 68, then $6^2 + 8^2 = 36 + 64 = 100, and finally $1^2 + 0^2 + 0^2 = 1. In the theory of linear recurrences, 82 is the sixth Pell–Lucas number (also known as a companion Pell number), which forms a sequence analogous to the Pell numbers but with initial terms 2 and 2, satisfying the recurrence P_n = 2P_{n-1} + P_{n-2}. The Pell–Lucas sequence begins 2, 2, 6, 14, 34, 82, 198, 478, and these numbers arise in solutions to Pell's equation and related Diophantine problems.^[14]^[15] Although 82 is a semiprime as noted in number theory, it also appears in combinatorial contexts near certain binomial coefficients, such as \binom{13}{2} = 78 or \binom{9}{3} = 84, highlighting its proximity to values in expansions of (1 + x)^n. In geometric and analytic settings, 82 degrees represents an acute angle, with \sin 82^\circ \approx 0.9903 and \tan 82^\circ \approx 7.1154, values that appear in trigonometric calculations but without a distinctive identity unique to 82.^[16]

In science

Chemistry

Lead is a chemical element with the atomic number 82 and the symbol Pb, derived from the Latin plumbum, the ancient name for lead, associated with its use in water pipes and possibly from a root meaning "to flow" due to its low melting point.^[1] It belongs to group 14 of the periodic table and is classified as a post-transition metal, characterized by its soft, malleable, and ductile nature.^[1] Lead exhibits a bluish-white luster when freshly cut but quickly tarnishes to a dull gray in air due to oxidation.^[1] Key physical properties of lead include a density of 11.342 g/cm³ at 20°C, making it one of the denser common metals, and a low melting point of 327.46°C, which contributes to its historical malleability for shaping.^[1] Its standard atomic mass is approximately 207.2 u, reflecting the weighted average of its naturally occurring isotopes.^[1] The electron configuration of lead is [Xe] 4f¹⁴ 5d¹⁰ 6s² 6p², with the outermost electrons in the 6s and 6p orbitals conferring its chemical reactivity, particularly in forming +2 and +4 oxidation states.^[17] Lead has four stable isotopes: ²⁰⁴Pb (1.4% abundance), ²⁰⁶Pb (24.1%), ²⁰⁷Pb (22.1%), and ²⁰⁸Pb (52.4%), the latter being the most abundant and an end product of the thorium-232 decay chain.^[1] Notably, ²⁰⁶Pb and ²⁰⁷Pb result from the decay of uranium-238 and uranium-235, respectively, enabling their use in uranium-lead radiometric dating to determine the age of ancient rocks and minerals with high precision, often via zircon crystals that incorporate uranium but exclude initial lead.^[18] This method relies on the known half-lives of uranium isotopes (4.47 billion years for U-238 and 704 million years for U-235) and ratios such as ²⁰⁶Pb/²⁰⁴Pb to correct for primordial lead.^[18] Despite its utility, lead is a potent cumulative neurotoxin and heavy metal poison that bioaccumulates in the body, causing neurological damage, developmental delays in children, and cardiovascular issues even at low exposure levels.^[19] Historically, it was widely used in ancient Rome for water pipes (plumbum), paints, cosmetics, and wine additives, leading to widespread chronic poisoning among elites and contributing to health epidemics like gout and sterility.^[19] In modern applications, lead persists in lead-acid batteries for vehicles and uninterruptible power supplies, as well as in radiation shielding for medical and nuclear facilities due to its high density and ability to absorb gamma rays.^[1] Regulatory efforts, such as the U.S. EPA's phase-out of leaded gasoline from 1973 to 1996, have reduced exposure, though legacy contamination remains a global concern.^[19]

Astronomy

In astronomy, the number 82 is associated with several notable cataloged objects, including galaxies and an asteroid. Messier 82 (M82), also known as NGC 3034, is an irregular galaxy located in the constellation Ursa Major, approximately 12 million light-years from Earth. It is renowned as a prototypical starburst galaxy, where intense star formation occurs at a rate estimated to be ten times that of the Milky Way, driven by gravitational interactions with its neighbor M81. M82's recession velocity relative to the Milky Way is about 203 km/s, indicating its motion within the local universe. One of its most significant events was the discovery of supernova SN 2014J on January 21, 2014, which was the closest type Ia supernova observed in the past four decades and provided valuable data for supernova research. Recent observations by the James Webb Space Telescope in 2024 and Hubble in 2025 have captured high-resolution images of M82's core, highlighting its intense star formation and galactic outflows.^[20] Asteroid (82) Alkmene, discovered on September 6, 1868, by astronomer Prosper Henry at the Paris Observatory, is a main-belt asteroid with an estimated diameter of about 62 kilometers. It is classified as an S-type asteroid, composed primarily of silicates and metals, and orbits the Sun at an average distance of 2.75 astronomical units with a period of roughly 4.57 years. Observations have revealed its irregular shape and low albedo, typical of many asteroids in the inner main belt.^[21]

In other fields

Sports

In American football, the number 82 has been prominently worn by several Hall of Fame-caliber players. Jason Witten, a tight end for the Dallas Cowboys from 2003 to 2017 and again from 2019 to 2020, donned #82 throughout his career, amassing 1,228 receptions for 13,046 yards and 74 touchdowns, establishing himself as one of the most durable and productive players at his position.^[22] Similarly, John Stallworth, a wide receiver for the Pittsburgh Steelers from 1974 to 1987, wore #82 and contributed significantly to four Super Bowl victories, recording 537 receptions for 8,723 yards and 63 touchdowns before his induction into the Pro Football Hall of Fame in 2002.^[23] In baseball, the number 82 is notably associated with Hall of Famer Kid Nichols through his managerial record rather than a jersey number, as uniform numbers were not used during his playing era. Nichols, a dominant pitcher from 1890 to 1906 who secured 361 wins—ranking seventh all-time—later managed the minor-league Kansas City Blue Stockings to an 82–54 record in 1899, securing the Western League pennant.^[24]^[25] In ice hockey, #82 has been worn by several NHL players, though none reach the iconic status of some lower numbers; notable examples include Martin Straka, who donned it for the Pittsburgh Penguins from 1992 to 1997 and later for other teams, accumulating 278 goals and 610 assists over 980 games.^[26] The number sees occasional use across the league but lacks a singular Hall of Fame-level figure. In soccer, jersey number 82 is rarely assigned to prominent players due to the preference for lower numbers, appearing sporadically among substitutes or in youth systems without association to any legendary athletes. Beyond jersey assignments, 82 holds structural significance in basketball as the standard number of regular-season games per team in the NBA, a format established since the 1967–68 season to balance competition and revenue. No major world records in professional sports are directly tied to the number 82.

Transportation and geography

U.S. Route 82 is a major east-west highway spanning approximately 1,609 miles (2,589 km) across the southern United States, connecting Alamogordo, New Mexico, to Brunswick, Georgia.^[27] It traverses six states: New Mexico, Texas, Arkansas, Mississippi, Alabama, and Georgia, providing a key corridor for freight and passenger travel through rural and urban areas in the region.^[28] Interstate 82, located in the Pacific Northwest, extends 144 miles (232 km) from its western terminus at Interstate 90 in Ellensburg, Washington, to its eastern end at Interstate 84 near Hermiston, Oregon.^[29] This interstate primarily runs through south-central Washington and a short segment in northeastern Oregon, serving agricultural areas and connecting to major trade routes along the Columbia River.^[29] In addition to federal highways, state and local routes numbered 82 exist elsewhere. Pennsylvania Route 82 is a 32-mile (51 km) north-south state highway in Chester County, Pennsylvania, running from the Delaware state line near Kennett Square to Elverson, primarily serving rural communities and connecting to U.S. Route 30.^[30] Across the Atlantic, Scotland's A82 is a trunk road stretching 167 miles (269 km) from Glasgow to Inverness, passing through the scenic Loch Lomond and Trossachs National Park and Glen Coe, functioning as a vital link for tourism and Highland access.^[31] Geographically, the 82nd meridian west of Greenwich is a line of longitude running from the North Pole through the Arctic Ocean, North America (including parts of Canada and the eastern United States), the Gulf of Mexico, the Caribbean, and South America to the South Pole.^[32] It notably crosses the Florida Panhandle and serves as a historical boundary reference in North American cartography. The 82nd meridian east similarly extends from pole to pole, traversing the Arctic Ocean, Asia (including Russia, India, and Indonesia), and the Indian and Southern Oceans.^[33] The 82nd parallel north, at 82 degrees latitude, forms a circle around the Earth entirely within the Arctic region, approximately 888 km (551 miles) south of the North Pole and well north of the Arctic Circle at 66.5 degrees.^[34] This latitude passes over the Arctic Ocean and fringes of northern Greenland and Svalbard, Norway, where it experiences extreme polar conditions with nearly continuous daylight in summer and darkness in winter.^[35]

References

[1]
Lead | Pb (Element) - PubChem - NIH
Lead is a chemical element with symbol Pb and atomic number 82. Classified as a post-transition metal, Lead is a solid at 25°C (room temperature).
[2]
Lead (Pb) Toxicty: What is Lead? | Environmental Medicine | ATSDR
Lead has an atomic number of 82 and an atomic weight of 207.2. It is a bluish-grey metal that tarnishes easily in air to a dark grey. The density of lead is ...
[3]
Learn about Lead | US EPA
Sep 9, 2025 · Lead is a naturally occurring element found in small amounts in the earth's crust. While it has some beneficial uses, it can be toxic to humans and animals, ...Missing: properties | Show results with:properties
[4]
Decimal to Binary
Decimal, Binary. 0, 0. 1, 1. 2, 10. 3, 11. 4, 100. 5, 101. 6, 110. 7, 111. 8, 1000. 9, 1001 ... 82, 1010010. 83, 1010011. 84, 1010100. 85, 1010101. 86, 1010110.
[5]
[PDF] ASCII TABLE - Decimal Hexadecimal Binary Octal Char
Decimal Hexadecimal Binary Octal Char. Decimal Hexadecimal Binary Octal ... 82. 52. 1010010 122. R. 35. 23. 100011 43. 83. 53. 1010011 123 S. 36. 24. 100100 44.
[6]
[PDF] Decimal/Hex/Binary Conversion Table
Decimal/Hex/Binary Conversion Table. Dec Hex Binary Dec Hex Binary Dec ... 18 12 00010010 82 52 01010010 146 92 10010010 210 D2 11010010. 19 13 00010011 ...
[7]
[PDF] EXPRESSNG NUMBERS IN DIFFERENT BASES
Sep 6, 2022 · convert(decimal form, base,3). This produces the table- decimal 1. 2. 3 ... So 82 decimal equals 10001 in ternary. Addition in ternary ...
[8]
Semiprime -- from Wolfram MathWorld
The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes) are 6, 10, 14, 15, 21, 22, 26, 33, 34, ... (OEIS A006881). The square of ...
[9]
A001358 - OEIS
Numbers of the form p*q where p and q are primes, not necessarily distinct. These numbers are sometimes called semiprimes or 2-almost primes.
[10]
A000010 - OEIS
A000010 - OEIS. Euler totient function phi(n): count numbers <= n and prime to n. Number of elements in a reduced residue system modulo n. a(n) equals the ...
[11]
A143757 - OEIS
Aliquot sequence starting at 82. 1. 82, 44, 40, 50, 43, 1, 0, 0, 0, 0, 0, 0 ... Mathar, Oct 11 2017. CROSSREFS. Sequence in context: A050678 A354501 A033402 * ...
[12]
A002203 - OEIS
Companion Pell numbers: a(n) = 2*a(n-1) + a(n-2), a(0) = a(1) = 2. (Formerly M0360 N0136). 168. 2, 2, 6, 14, 34, 82, 198, 478 ...
[13]
Pell-Lucas Number -- from Wolfram MathWorld
The Pell-Lucas numbers are the V_n s in the Lucas sequence with P=2 and Q=-1, and correspond to the Pell-Lucas polynomial Q_n(1).Missing: properties | Show results with:properties
[14]
Find Value of Tan 82 Degrees - Cuemath
The tan of 82 degrees equals the y-coordinate(0.9903) divided by x-coordinate(0.1392) of the point of intersection (0.1392, 0.9903) of unit circle and r.
[15]
Lead - Element information, properties and uses | Periodic Table
Lead, atomic number 82, is a solid with a melting point of 327.462°C. It has been mined for over 6,000 years and is slow, dull, and heavy.
[16]
Clocks in the Rocks
### Summary of Uranium-Lead Dating Using Lead Isotopes
[17]
Lead Poisoning: A Historical Perspective | About EPA
Sep 16, 2016 · What they did not realize was that their everyday low-level exposure to the metal rendered them vulnerable to chronic lead poisoning, even while ...
[18]
Jason Witten Stats, Height, Weight, Position, Draft, College
Jason Witten had 1,228 receptions over his career. How many receiving yards did Jason Witten have? Jason Witten had 13,046 receiving yards over his career.
[19]
John Stallworth Stats, Height, Weight, Position, Draft, College
How many receptions did John Stallworth have? John Stallworth had 537 receptions over his career. How many receiving yards did John Stallworth have? John ...
[20]
Kid Nichols - BR Bullpen - Baseball-Reference.com
May 4, 2025 · He was 26-7 in his first year there, and the club went on to win the league crown. The next summer, he posted a 21-12 mark.
[21]
https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=82
[22]
End of US highway 82
Nov 28, 2024 · US route 82 has endpoints in Brunswick (Georgia) and Alamogordo (New Mexico), with historic termini in Midway, Las Cruces, Lubbock, ...
[23]
U.S. Route 82 - AARoads Wiki
Nov 18, 2023 · US 82 is a 232-mile-long (373 km) U.S. Highway in the U.S. state of Georgia. ... Click for the article on the U.S. Route shield United States ...
[24]
Interstate 82
I-82 connects I-90 at Ellensburg, Washington with I-84 south of Hermiston, Oregon. I-82 also serves the Tri-City area in southeast Washington.
[25]
Route 82 Pennsylvania - AARoads
Feb 3, 2021 · Pennsylvania Route 82 travels north from the Delaware state line to Kennett Square, Coatesville and Elverson in Chester County.
[26]
A82 improvements - Transport Scotland
The A82 is a vital artery for communities in the western Highlands, linking Fort William, the Highlands and the Western Isles with Glasgow and the Central ...
[27]
GPS coordinates of 82nd meridian west. Latitude: 90.0000 Longitude
The meridian 82° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, North America, the Gulf of Mexico, ...Missing: geography | Show results with:geography<|separator|>
[28]
GPS coordinates of 82nd meridian east. Latitude: 90.0000 Longitude
The meridian 82° east of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Asia, the Indian Ocean, the Southern ...Missing: geography | Show results with:geography
[29]
82 degrees north - Why Evolution Is True
Jul 12, 2025 · Each line of latitude is 111 km apart, so 8 degrees of latitude from 82 degrees to the pole at 90 degrees is 888 km from the pole. A lot of ...
[30]
GPS coordinates of 82nd parallel north. Latitude: 82.0000 Longitude
Satellite map of 82nd parallel north. The 82nd parallel north is a circle of latitude that is 82 degrees north of the Earth's equatorial plane, in the Arctic.