subclass_of giuga number
In ___MASK55___ theory , the prime factors of a positive integer are the prime numbers that divide that integer exactly . The prime factorization of a positive integer is a list of the integer ' s prime factors , together with their multiplicities ; the process of determining these factors is called integer factorization . The fundamental theorem of arithmetic says that every positive integer has a single unique prime factorization . A prime ___MASK55___ ( or a prime ) is a ___MASK91___ greater than 1 that has no positive divisors other than 1 and itself . A ___MASK91___ greater than 1 that is not a prime ___MASK55___ is called a composite ___MASK55___ . 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 ___MASK55___ 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 . A Giuga ___MASK55___ is a composite ___MASK55___ n such that for each of its distinct prime factors pi we have p_i ( ( n \ over p_i ) - 1 ) , or equivalently such that for each of its distinct prime factors pi we have p_i ^ 2 ( n - p_i ) . The Giuga numbers are named after the mathematician Giuseppe Giuga , and relate to his conjecture on primality . A composite ___MASK55___ is a positive integer that can be formed by multiplying together two smaller positive integers . Equivalently , it is a positive integer that has at least one divisor other than 1 and itself . Every positive integer is composite , prime , or the unit 1 , so the composite numbers are exactly the numbers that are not prime and not a unit .