subclass_of leyland number
An ___MASK2___ ( from the ___MASK10___ " ___MASK2___ " meaning " whole ") is a ___MASK63___ that can be written without a fractional component . For example , 21 , 4 , 0 , and  2048 are integers , while 9 . 75 , , and are not . In ___MASK63___ theory , a Leyland ___MASK63___ is a ___MASK63___ 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 ___MASK63___ 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 ___MASK63___ 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 . In ___MASK77___ , a ___MASK55___ 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 to 1 , every integer is a ___MASK55___ . 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 ". ___MASK63___ theory or , in older usage , arithmetic is a branch of pure mathematics devoted primarily to the ___MASK58___ of the integers . It is sometimes called " The Queen of Mathematics " because of its foundational place in the discipline . ___MASK63___ theorists ___MASK58___ 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 ). In ___MASK63___ theory , integer factorization is the decomposition of a composite ___MASK63___ into a product of smaller integers . If these integers are further restricted to prime numbers , the process is called prime factorization . A prime ___MASK63___ ( or a prime ) is a ___MASK3___ greater than 1 that has no positive divisors other than 1 and itself . A ___MASK3___ greater than 1 that is not a prime ___MASK63___ is called a composite ___MASK63___ . 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 ___MASK63___ 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 . ___MASK10___ ( ___MASK10___ : ) is a classical language belonging to the Italic branch of the Indo - European languages . The ___MASK52___ is derived from the Etruscan and Greek alphabets . In ___MASK25___ theory , an algebraic integer is a ___MASK20___ 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 ___MASK42___ is the integral ___MASK30___ of regular integers formula_1 in complex numbers . Paul Leyland is a British ___MASK63___ theorist who has studied integer factorization and primality testing . A primality ___MASK15___ is an ___MASK33___ 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 ___MASK51___ , whereas primality testing is comparatively easy ( its running ___MASK72___ is polynomial in the size of the input ). Some primality tests " prove " that a number is prime , while others like Miller  Rabin prove that a number is composite . Therefore , the latter might be called " compositeness tests " instead of primality tests . Broadly speaking , pure ___MASK77___ is ___MASK77___ that studies entirely abstract concepts . This was a recognizable ___MASK75___ 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 .