What Is a Prime Number?
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. In other words, it cannot be divided evenly by any other whole number.
The first few primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
Notice that 2 is the only even prime number — every other even number can be divided by 2, so it has more than two factors. The number 1 is not considered prime because it has only one factor (itself), which would break many important mathematical rules if it were included.
What Makes a Number NOT Prime?
Numbers that have more than two factors are called composite numbers. For example:
- 6 = 2 × 3 (factors: 1, 2, 3, 6) — composite
- 15 = 3 × 5 (factors: 1, 3, 5, 15) — composite
- 7 = only 1 × 7 — prime
The Fundamental Theorem of Arithmetic
Here is what makes primes truly special: every whole number greater than 1 can be written as a unique product of prime numbers. This is called the Fundamental Theorem of Arithmetic.
For example:
- 12 = 2 × 2 × 3
- 100 = 2 × 2 × 5 × 5
- 360 = 2 × 2 × 2 × 3 × 3 × 5
Primes are, in this sense, the "atoms" of all numbers — every integer is built from them in exactly one way.
How to Check If a Number Is Prime
To test whether a number n is prime, you only need to check whether it's divisible by any prime number up to its square root (√n). If none of them divide evenly into n, it's prime.
- Check if the number is divisible by 2 (is it even?)
- Check divisibility by 3, 5, 7, 11… up to √n
- If none divide evenly, the number is prime
For example, to check if 97 is prime: √97 ≈ 9.8, so we only need to test primes up to 9 (2, 3, 5, 7). None of them divide 97 evenly, so 97 is prime.
Are There Infinitely Many Primes?
Yes — and this was proven by the ancient Greek mathematician Euclid around 300 BCE in one of mathematics' most elegant proofs. The argument goes like this: assume there is a finite list of all primes. Multiply them all together and add 1. This new number either is prime (contradicting our "complete" list) or has a prime factor not in our list. Either way, our list was incomplete — proving primes go on forever.
Primes and Internet Security
Prime numbers are not just abstract curiosities — they underpin modern cybersecurity. The encryption system used to secure online banking, emails, and websites (called RSA encryption) relies on the fact that while multiplying two large prime numbers together is easy, factoring the result back into its prime components is computationally extremely difficult.
When you see "https" in a web address, you are benefiting from the mathematical properties of prime numbers.
The Unsolved Mystery of Primes
Despite millennia of study, primes still hold deep mysteries. The Riemann Hypothesis — one of mathematics' greatest unsolved problems — concerns the distribution of prime numbers. It has remained unproven since 1859 and carries a $1 million prize for whoever solves it.
Prime numbers remind us that even the most basic mathematical objects can lead to questions that challenge the greatest minds in history.