239 (two hundred ndthirty-nine) is the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

following 238 and preceding

240 __NOTOC__ Year 240 ( CCXL) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sabinus and Venustus (or, less frequently, year 993 ''Ab u ...

In mathematics

It is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

. The next is 241, with which it forms a pair of

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...

s; hence, it is also a

Chen prime A prime number ''p'' is called a Chen prime if ''p'' + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2''p'' + 2 therefore satisfies Chen's theorem. The Chen primes are named after Chen Jingru ...

. 239 is a

Sophie Germain prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + ...

and a Newman–Shanks–Williams prime. It is an

Eisenstein prime In mathematics, an Eisenstein prime is an Eisenstein integer : z = a + b\,\omega, \quad \text \quad \omega = e^, that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units , itself ...

with no imaginary part and real part of the form 3''n'' − 1 (with no exponentiation implied). 239 is also a

happy number In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy number because ...

. 239 is the smallest positive integer ''d'' such that the imaginary

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...

Q() has class number = 15.

HAKMEM HAKMEM, alternatively known as AI Memo 239, is a February 1972 "memo" (technical report) of the MIT AI Lab containing a wide variety of hacks, including useful and clever algorithms for mathematical computation, some number theory and schematic ...

(incidentally AI memo 239 of the

MIT AI Lab Computer Science and Artificial Intelligence Laboratory (CSAIL) is a research institute at the Massachusetts Institute of Technology (MIT) formed by the 2003 merger of the Laboratory for Computer Science (LCS) and the Artificial Intelligence Lab ...

) included an item on the properties of 239, including these: * When expressing 239 as a sum of

square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...

s, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positive

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

s (23 is the only other such integer), and the maximum number (19) of fourth powers. * 239/

169 Year 169 ( CLXIX) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Senecio and Apollinaris (or, less frequently, year 922 ''Ab urbe co ...

is a convergent of the

continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

of the

square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...

, so that 239² = 2 · 169² − 1. * Related to the above, = 45 °. * 239 · 4649 = 1111111, so 1/239 = 0.0041841 repeating, with period 7. * 239 can be written as ''b''^''n'' − ''b''^''m'' − 1 for ''b'' = 2, 3, and 4, a fact evidenced by its binary representation 11101111, ternary representation 22212, and

quaternary The Quaternary ( ) is the current and most recent of the three periods of the Cenozoic Era in the geologic time scale of the International Commission on Stratigraphy (ICS). It follows the Neogene Period and spans from 2.58 million year ...

representation 3233. * There are 239 primes < 1500. * 239 is the largest integer ''n'' whose

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...

can be written as the product of distinct factors between ''n'' + 1 and 2''n'', both included. * The only solutions of the Diophantine equation ''y''² + 1 = 2''x''⁴ in positive integers are (''x'', ''y'') = (1, 1) or (13, 239).

In other fields

239 is also: * K239 is

Mozart Wolfgang Amadeus Mozart (27 January 17565 December 1791), baptised as Joannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the Classical period. Despite his short life, his rapid pace of composition r ...

's only work for two

orchestra An orchestra (; ) is a large instrumental ensemble typical of classical music, which combines instruments from different families. There are typically four main sections of instruments: * bowed string instruments, such as the violin, viola, c ...

s. * 239 is the

atomic mass number The mass number (symbol ''A'', from the German word ''Atomgewicht'' tomic weight, also called atomic mass number or nucleon number, is the total number of protons and neutrons (together known as nucleons) in an atomic nucleus. It is approxi ...

of the most common isotope of

plutonium Plutonium is a radioactive chemical element with the symbol Pu and atomic number 94. It is an actinide metal of silvery-gray appearance that tarnishes when exposed to air, and forms a dull coating when oxidized. The element normally exh ...

Pu-239 Plutonium-239 (239Pu or Pu-239) is an isotope of plutonium. Plutonium-239 is the primary fissile isotope used for the production of nuclear weapons, although uranium-235 is also used for that purpose. Plutonium-239 is also one of the three main ...

* 239 is an area code representing part of southwest

Florida Florida is a state located in the Southeastern region of the United States. Florida is bordered to the west by the Gulf of Mexico, to the northwest by Alabama, to the north by Georgia, to the east by the Bahamas and Atlantic Ocean, and ...

in the

United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...

*239 is a world-famous lyceum of physics in mathematics in

Saint-Petersburg Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...

Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-ei ...

. * In ''

The Simpsons ''The Simpsons'' is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a satirical depiction of American life, epitomized by the Simpson family, which consists of Homer, Marge, Bart, ...

'' episode "

Homer's Night Out "Homer's Night Out" is the tenth episode of the American animated television series ''The Simpsons''. It originally aired on the Fox network in the United States on March 25, 1990. It was written by Jon Vitti and directed by Rich Moore.

Homer Homer (; grc, Ὅμηρος , ''Hómēros'') (born ) was a Greek poet who is credited as the author of the ''Iliad'' and the ''Odyssey'', two epic poems that are foundational works of ancient Greek literature. Homer is considered one of the ...

weighs himself and resolves to exercise. Six months later he weighs himself again and again resolves to exercise. Both times he weighed exactly 239 pounds. His weight was also seen in a daydream in the episode " King-Size Homer" *239 is the number of chapters in the

Book of Mormon The Book of Mormon is a religious text of the Latter Day Saint movement, which, according to Latter Day Saint theology, contains writings of ancient prophets who lived on the American continent from 600 BC to AD 421 and during an interlude ...

References

{{DEFAULTSORT:239 (Number) Integers