Prime Numbers

Intuition and core definition

A prime number is a whole number greater than $1$ with exactly two positive factors: $1$ and itself. A composite number greater than $1$ has more than two positive factors. The number $1$ is neither because it has only one positive factor; $2$ is the only even prime.

Notation, language, and conditions

For an integer $n>1$, it is enough to test prime divisors up to $\sqrt n$. If $n=ab$ is composite, at least one factor must be at most $\sqrt n$. A proof that a number is prime must rule out every possible prime divisor in that range, not merely show that several small numbers fail.

Why this idea matters

Prime numbers have exactly two positive factors and act as the irreducible pieces from which every positive integer above one is built.

A dependable method

1. Handle boundary cases: numbers at most $1$ are not prime.
2. Check divisibility by $2,3,5,7,\ldots$ up to $\sqrt n$.
3. If any division has remainder zero, display the nontrivial factor pair and classify the number as composite.
4. If none of those primes divides, classify it as prime.
5. For a range, use a sieve to cross out multiples rather than retesting each number independently.

Worked example

Common mistakes