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 numbers like $12$ are composite.

If a whole number greater than $1$ has more than two positive divisors, it is called composite. For example, $12$ is composite because it is divisible by $1, 2, 3, 4, 6,$ and $12$.

Prime Numbers Up To 50

Here are the prime numbers up to $50$:

$$2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29,\ 31,\ 37,\ 41,\ 43,\ 47$$

There is no simple repeating gap pattern. For example, $11$ and $13$ are close together, but the next gap from $23$ to $29$ is larger.

What Makes A Number Prime?

To be prime, a number must pass both conditions:

1. It must be greater than $1$.
2. Its only positive divisors must be $1$ and the number itself.

That is why $1$ is not prime, and it is also why $2$ is prime even though it is even. The number $2$ has exactly two positive divisors: $1$ and $2$.

How To Tell If A Number Is Prime

For a whole number $n > 1$, you can test whether it is prime by checking whether any whole number from $2$ up to $\lfloor \sqrt{n} \rfloor$ divides it exactly.

The reason is practical: if $n = ab$, then one of the factors must be less than or equal to $\sqrt{n}$. So if no divisor appears by the time you reach $\sqrt{n}$, there is no hidden larger-factor pair waiting above it.

In everyday work, people usually check small divisibility rules first:

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 $5$.

Those shortcuts do not prove a number is prime on their own, but they help rule out many composite numbers quickly.

Worked Example: Is $29$ Prime?

To test $29$, first note that

$$\sqrt{29} \approx 5.38$$

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

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

Checking $4$ adds nothing here because any multiple of $4$ is even, and $29$ is already not divisible by $2$.

No divisor up to $5$ works, so $29$ is prime.

Common Mistakes With Prime Numbers

Saying $1$ is prime

It is not. The definition requires exactly two positive divisors, and $1$ has only one.

Thinking every odd number is prime

Many odd numbers are composite. For example, $21$ is odd, but

$$21 = 3 \times 7$$

so it is not prime.

Checking too far

If you are only testing primality, you do not need to try every number less than $n$. Stopping at $\sqrt{n}$ is enough.

Where Prime Numbers Are Used

Prime numbers show up in factorization, divisibility, greatest common divisor problems, and least common multiple problems. They matter because every integer greater than $1$ can be broken into prime factors in a way that is unique up to order.

They also appear in modular arithmetic and cryptography. In cryptography, the setting is much more specialized, and large primes are used together with additional rules and algorithms.

Try A Similar Problem

Test $47$ and $51$ with the same square-root method. One is prime and one is composite, so this is a quick way to check whether the stopping rule at $\sqrt{n}$ makes sense to you.