Prime Numbers — List, Definition & How to Find Them

A prime number is a whole number greater than $1$ with exactly two positive divisors: $1$ and itself. So $2, 3, 5,$ and $7$ are prime, $1$ is not prime, and a number like $12$ is composite. A composite number is any whole number greater than $1$ with more than two positive divisors — $12$ is composite because it is divisible by $1, 2, 3, 4, 6,$ and $12$.

For reference, the primes up to $50$ are

There is no simple repeating gap pattern: $11$ and $13$ are close, but the jump from $23$ to $29$ is larger.

When You Run a Primality Test

You apply the test below whenever you need to decide whether a single whole number $n > 1$ is prime — in factorization, divisibility, GCD and LCM work, modular arithmetic, and (with much larger numbers and extra algorithms) cryptography. To be prime, a number must clear two conditions: it must be greater than $1$, and its only positive divisors must be $1$ and itself. That is why $1$ is not prime, and why $2$ is prime even though it is even — it has exactly two divisors, $1$ and $2$.

The Steps

For a whole number $n > 1$, test whether any whole number from $2$ up to $\lfloor \sqrt{n} \rfloor$ divides it exactly. The reason you can stop at the square root: if $n = ab$, then one factor must be $\le \sqrt{n}$, so if no divisor appears by then, no hidden larger-factor pair waits above it.

In practice, start with quick divisibility rules to eliminate easy cases:

1. If $n$ is even and greater than $2$, it is not prime.
2. If the digits add to a multiple of $3$, then $n$ is divisible by $3$.
3. If $n$ ends in $0$ or $5$ and is greater than $5$, it is divisible by $5$.

These shortcuts do not prove primality on their own, but they rule out many composites fast.

Full Worked Example: Is $29$ Prime?

First find the cutoff:

so it is enough to check whole-number divisors up to $5$.

• $29$ is not divisible by $2$ — it is odd.
• $29$ is not divisible by $3$ — $2 + 9 = 11$, not a multiple of $3$.
• $29$ is not divisible by $5$ — it does not end in $0$ or $5$.

Checking $4$ adds nothing, since any multiple of $4$ is even and $29$ already failed the $2$ test. No divisor up to $5$ works, so $29$ is prime.

Where Students Get Stuck (and How to Check)

• Saying $1$ is prime. It is not — the definition needs exactly two positive divisors, and $1$ has only one.
• Thinking every odd number is prime. Many odd numbers are composite; for instance $21 = 3 \times 7$. Always run the divisor check, do not assume.
• Checking too far. For primality you do not need every number below $n$ — stopping at $\sqrt{n}$ is enough. If you are testing past the square root, you are wasting effort.

Try It Yourself

Test $47$ and $51$ with the same square-root method. One is prime and one is composite, which is a quick way to confirm that the stopping rule at $\sqrt{n}$ truly makes sense to you. Prime numbers matter because every integer greater than $1$ factors into primes in a way that is unique up to order — the backbone of much of number theory.