nonprime

Nonprime is a term used in number theory to refer to an integer that is not a prime number. Primes are defined as natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself. By contrast, a nonprime is any integer that does not meet this criterion. In standard discussions of positive integers, the nonprimes greater than 1 are precisely the composite numbers, while the integers 1, 0, and negative numbers are not considered primes and are often described as neither prime nor composite.

Among positive integers, composite numbers have more than two positive divisors and can be factored into a product of smaller integers greater than 1. Every composite number can be expressed as a product of primes, and this prime factorization is unique up to the order of the factors (the fundamental theorem of arithmetic). This property underpins many algorithms for primality testing and integer factorization.

The term nonprime is less commonly used in formal contexts; mathematicians typically refer to nonprime integers as composite numbers when the discussion is restricted to positive integers greater than 1. In broader usage, nonprime may include 0 and negative integers, but these are not classified as primes under the standard definitions.

Nonprimes play a central role in topics such as divisibility, factorization, and the study of arithmetic functions. They form the complement of the prime numbers within the set of integers and are essential for understanding the structure of the integers under multiplication.