nonprimes

Nonprime is a term used in number theory to refer to integers that are not prime. When the discussion is restricted to positive integers, this category includes 1 and all composite numbers; in many contexts, nonprime is taken to mean the positive integers greater than 1 that are not prime, i.e., the composite numbers.

Examples of nonprimes include 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, and so on. In contrast, prime numbers are the positive integers greater than 1 with exactly two distinct divisors: 1 and the number itself.

A positive integer n > 1 is nonprime if and only if it has a nontrivial divisor, that is, a divisor other than 1 and n. Equivalently, n is nonprime if it is divisible by some prime number p with p ≤ sqrt(n). This characterization underpins many primality tests and factorization methods.

In practice, the nonprimes (aside from 1) are precisely the composite numbers. The primes and composites partition the positive integers greater than 1. Regarding distribution, primes become sparser as numbers grow, while nonprimes become more prevalent; the prime number theorem shows that the proportion of primes near x is about 1/log x, which tends to zero as x grows.

Nonprimes are central to algorithms such as the Sieve of Eratosthenes, which marks nonprimes to identify primes. They also play a key role in factorization, divisibility tests, and various problems in elementary and analytic number theory. See also primes, composite numbers, and sieve methods.