TASK DEFINITION: In this task, you are given a context, a subject, a relation, and many options. Based on the context, from the options select the object entity that has the given relation with the subject. Answer with text (not indexes).
PROBLEM: Context: Latin (Latin: ) is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets., In mathematics, a rational number is any number that can be expressed as the quotient or fraction "p"/"q" of two integers, a numerator "p" and a non-zero denominator "q". Since "q" may be equal to1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", is usually denoted by a boldface Q (or blackboard bold formula_1, Unicode ); it was thus denoted in 1895 by Giuseppe Peano after "quoziente", Italian for "quotient"., In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to prime numbers, the process is called prime factorization., A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests "prove" that a number is prime, while others like MillerRabin prove that a number is composite. Therefore, the latter might be called "compositeness tests" instead of primality tests., Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers)., A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 3, 1 1 3, etc. are all valid factorizations of 3., Paul Leyland is a British number theorist who has studied integer factorization and primality testing., Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. This was a recognizable category of mathematical activity from the 19th century onwards, at variance with the trend towards meeting the needs of navigation, astronomy, physics, economics, engineering, and so on., In number theory , a Leyland number is a number of the form x ^ y + y ^ x where x and y are integers greater than 1 . They are named after the mathematician Paul Leyland . The first few Leyland numbers are 8 , 17 , 32 , 54 , 57 , 100 , 145 , 177 , 320 , 368 , 512 , 593 , 945 , 1124 ( sequence A076980 in OEIS ) . The requirement that x and y both be greater than 1 is important , since without it every positive integer would be a Leyland number of the form x1 + 1x . Also , because of the commutative property of addition , the condition x y is usually added to avoid double - covering the set of Leyland numbers ( so we have 1 < y x ) . The first prime Leyland numbers are 17 , 593 , 32993 , 2097593 , 8589935681 , 59604644783353249 , 523347633027360537213687137 , 43143988327398957279342419750374600193 ( A094133 ) corresponding to 32 +23 , 92 +29 , 152 +215 , 212 +221 , 332 +233 , 245 +524 , 563 +356 , 3215 +1532 . One can also fix the value of y and consider the sequence of x values that gives Leyland primes , for example x2 + 2x is prime for x = 3 , 9 , 15 , 21 , 33 , 2007 , 2127 , 3759 , ... ( A064539 ) . By November 2012 , the largest Leyland number that had been proven to be prime was 51226753 + 67535122 with 25050 digits . From January 2011 to April 2011 , it was the largest prime whose primality was proved by elliptic curve primality proving . In December 2012 , this was improved by proving the primality of the two numbers 311063 + 633110 ( 5596 digits ) and 86562929 + 29298656 ( 30008 digits ) , the latter of which surpassed the previous record . There are many larger known probable primes such as 3147389 + 9314738 , but it is hard to prove primality of large Leyland numbers . Paul Leyland writes on his website : '' More recently still , it was realized that numbers of this form are ideal test cases for general purpose primality proving programs . They have a simple algebraic description but no obvious cyclotomic properties which special purpose algorithms can exploit . '' There is a project called XYYXF to factor composite Leyland numbers ., An integer (from the Latin "integer" meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and2048 are integers, while 9.75, , and are not., In Algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in formula_1 (the set of integers). The set of all algebraic integers, , is closed under addition and multiplication and therefore is a commutative subring of the complex numbers. The ring is the integral closure of regular integers formula_1 in complex numbers., Subject: leyland number, Relation: subclass_of, Options: (A) algorithm (B) blackboard (C) branch (D) category (E) complex number (F) decomposition (G) economics (H) engineering (I) integer (J) language (K) latin (L) mathematics (M) meeting (N) natural (O) natural number (P) number (Q) out (R) polynomial (S) problem (T) process (U) rational number (V) ring (W) set (X) test (Y) theorem (Z) theory ([) thought (\) time

SOLUTION: natural number

PROBLEM: Context: Rupert James Hector Everett (born 29 May 1959) is an English actor and writer. He first came to public attention in 1981, when he was cast in Julian Mitchell's play and subsequent film "Another Country" (1984) as an openly gay pupil at an English public school in the 1930s; the role earned him his first BAFTA Award nomination. He went on to receive a second BAFTA Award nomination and his first Golden Globe Award nomination for his role in "My Best Friend's Wedding" (1997), followed by a second Golden Globe nomination for "An Ideal Husband" (1999)., Dance with a Stranger is a 1985 British drama film, directed by Mike Newell. Telling the story of Ruth Ellis, the last woman to be hanged in Britain in the 1950s, this moving biographical British film won critical acclaim, and brought particular notice to the careers of both Miranda Richardson and Rupert Everett. The screenplay was by Shelagh Delaney, author of "A Taste of Honey", and her third major screenplay. The story of Ellis, which this film dramatises, has very considerable resonance in Britain since it provided part of the background to the long-term national debates which led to the progressive abolition of capital punishment from 1965 on., Ruth Ellis (9 October 1926 13 July 1955) was the last woman to be executed in the United Kingdom, after being convicted of the murder of her lover, David Blakely., Miranda Jane Richardson ( born 3 March 1958 ) is an English stage , film and television actress . She made her film debut playing Ruth Ellis in Dance with a Stranger in 1985 , and went on to receive Academy Award nominations for Damage ( 1992 ) and Tom & Viv ( 1994 ) . For Damage , she won the BAFTA Award for Best Supporting Actress . She has also won Golden Globe Awards for Enchanted April ( 1992 ) and the TV film Fatherland ( 1994 ) . Richardson began her career on stage and made her West End debut in the 1980 play Moving . She received a Best Actress Olivier Award nomination for the 1987 Royal Court production of A Lie of the Mind . On television , she has starred in Blackadder ( 1986 - 89 ) , A Dance to the Music of Time ( 1997 ) , Merlin ( 1998 ) , The Lost Prince ( 2003 ) and the sitcom The Life and Times of Vivienne Vyle ( 2007 ) . Her other films include Empire of the Sun ( 1987 ) , The Crying Game ( 1992 ) , The Apostle ( 1997 ) , Sleepy Hollow ( 1999 ) , Spider ( 2002 ) , Harry Potter and the Goblet of Fire ( 2005 ) and Made in Dagenham ( 2010 ) ., Subject: miranda richardson, Relation: occupation, Options: (A) actor (B) actress (C) author (D) major (E) television (F) united kingdom

SOLUTION: actor

PROBLEM: Context: Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a giant planet with a mass one-thousandth that of the Sun, but two and a half times that of all the other planets in the Solar System combined. Jupiter is a gas giant, along with Saturn, with the other two giant planets, Uranus and Neptune, being ice giants. Jupiter was known to astronomers of ancient times. The Romans named it after their god Jupiter. When viewed from Earth, Jupiter can reach an apparent magnitude of 2.94, bright enough for its reflected light to cast shadows, and making it on average the third-brightest object in the night sky after the Moon and Venus., An astronaut or cosmonaut is a person trained by a human spaceflight program to command, pilot, or serve as a crew member of a spacecraft. Although generally reserved for professional space travelers, the terms are sometimes applied to anyone who travels into space, including scientists, politicians, journalists, and tourists., Buzz Aldrin ( born Edwin Eugene Aldrin , Jr. , January 20 , 1930 ) is an American engineer and former astronaut , and the second person to walk on the Moon . He was the Lunar Module Pilot on Apollo 11 , the first manned lunar landing in history . He set foot on the Moon at 03:15:16 on July 21 , 1969 ( UTC ) , following mission commander Neil Armstrong . He is a former U.S. Air Force officer and a Command Pilot ., The Korean War (in South Korean , "Korean War"; in North Korean , "Fatherland Liberation War"; 25 June 1950 27 July 1953) began when North Korea invaded South Korea. The United Nations, with the United States as the principal force, came to the aid of South Korea. China came to the aid of North Korea, and the Soviet Union gave some assistance., The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets and small Solar System bodies. Of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury., Purdue University is a public research university located in West Lafayette, Indiana and is the main campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money to establish a college of science, technology, and agriculture in his name. The first classes were held on September 16, 1874, with six instructors and 39 students., The Moon is an astronomical body that orbits planet Earth, being Earth's only permanent natural satellite. It is the fifth-largest natural satellite in the Solar System, and the largest among planetary satellites relative to the size of the planet that it orbits (its primary). Following Jupiter's satellite Io, the Moon is second-densest satellite among those whose densities are known. , Neil Alden Armstrong (August 5, 1930 August 25, 2012) was an American astronaut and the first person to walk on the Moon. He was also an aerospace engineer, naval aviator, test pilot, and university professor. Before becoming an astronaut, Armstrong was an officer in the U.S. Navy and served in the Korean War. After the war, he earned his bachelor's degree at Purdue University and served as a test pilot at the National Advisory Committee for Aeronautics (NACA) High-Speed Flight Station, where he logged over 900 flights. He later completed graduate studies at the University of Southern California., A spacecraft is a vehicle, or machine designed to fly in outer space. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, space colonization, planetary exploration, and transportation of humans and cargo., The University of Southern California (USC or SC) is a private research university founded in 1880 with its main campus in Los Angeles, California. As California's oldest private research university, USC has historically educated a large number of the region's business leaders and professionals. In recent decades, the university has also leveraged its location in Los Angeles to establish relationships with research and cultural institutions throughout Asia and the Pacific Rim. An engine for economic activity, USC contributes approximately $5 billion annually to the economy of the Los Angeles metropolitan area., The National Advisory Committee for Aeronautics (NACA) was a U.S. federal agency founded on March 3, 1915, to undertake, promote, and institutionalize aeronautical research. On October 1, 1958, the agency was dissolved, and its assets and personnel transferred to the newly created National Aeronautics and Space Administration (NASA). NACA was pronounced as discrete letters, rather than as a whole word (and after NASA first was established, its acronym was pronounced as discrete letters in the early years)., Subject: buzz aldrin, Relation: employer, Options: (A) exploration (B) national advisory committee for aeronautics (C) national aeronautics and space administration (D) professor (E) purdue university (F) the sun (G) united nations (H) university of southern california

SOLUTION:
national aeronautics and space administration