The standard prime number definition is a whole number greater than 1 that can be divided evenly only by 1 and by itself. So 7 is prime, because nothing except 1 and 7 goes into it without a remainder, while 8 is not, because 2 and 4 also divide it.
Numbers above 1 that are not prime are called composite. Every composite number can be built by multiplying primes together, which is why mathematicians call primes the atoms of arithmetic.
Entry at a glance
Meaning
Whole number above 1 with exactly two factors: 1 and itselfPart of speech
Noun phrase; prime alone can be an adjective or nounPronunciation
PRYM NUM-berFirst primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29Opposite
Composite numberSpecial cases
1 is neither prime nor composite; 2 is the only even primeOrigin of prime
Latin primus, first
Applying the prime number definition step by step
To check the meaning against a number, ask two questions:
- Is it a whole number bigger than 1?
- Does it have exactly two different factors, 1 and itself?
If both answers are yes, it is prime. Take 13: its only factors are 1 and 13, so it passes. Take 15: it also divides by 3 and 5, so it is composite. Take 1: it has only one factor, itself, so it fails the second test and is not prime.
A factor (or divisor) is a number that divides another with no remainder. Primes are the numbers with the fewest possible factors.
The primes under 100
| Range | Primes |
|---|---|
| 1 to 20 | 2, 3, 5, 7, 11, 13, 17, 19 |
| 21 to 40 | 23, 29, 31, 37 |
| 41 to 60 | 41, 43, 47, 53, 59 |
| 61 to 80 | 61, 67, 71, 73, 79 |
| 81 to 100 | 83, 89, 97 |
Notice they get a little rarer as numbers grow, yet they never stop. Over two thousand years ago, Greek mathematicians had already proved that there are infinitely many primes.
A quick way to test a number
You only need to try dividing by primes up to the square root of the number. If none divides it, the number is prime.
The sieve of Eratosthenes, named after an ancient Greek scholar, finds all primes up to a limit: list the numbers, then cross out every multiple of 2, then of 3, then of 5, and so on. Whatever survives is prime. Few methods are older; it counts among the earliest algorithms.
Why 1 is not a prime number
Students ask about 1 more than any other number. Under the modern definition a prime needs exactly two distinct factors, and 1 has only one. There is also a deeper reason: every whole number above 1 can be written as a product of primes in only one way, ignoring order (12 is 2 times 2 times 3). This rule, called the fundamental theorem of arithmetic, would break if 1 counted, because you could add as many 1s as you liked. Older writers sometimes did list 1 as prime, but mathematicians settled on excluding it.
Using the term in sentences
- Our maths teacher asked us to circle every prime number on the hundred chart.
- Since 29 is a prime number, the class of 29 students could not be split into equal teams.
- Is 51 a prime number? No, it equals 3 times 17.
- The code lock uses the product of two very large prime numbers.
Where primes matter outside class
Large primes protect much of online banking and messaging. Some encryption systems rely on the fact that multiplying two huge primes is easy, but working backward to find them from the product is extremely slow. Primes also appear in nature puzzles: some cicada species emerge every 13 or 17 years, and scientists have suggested prime cycles may help them avoid predators, though this is still debated.
Related science vocabulary lives in science and ideas; also look at abstract for how maths ideas are described.
Frequently asked questions
What is the definition of prime number in simple words?
A prime is a whole number bigger than 1 that only 1 and the number itself divide without a remainder. Seven sweets cannot be shared equally among 2, 3, 4, 5 or 6 children, which makes the idea easy to see.
Is 2 a prime number?
Yes. It is the smallest prime and the only even one, because every other even number can also be divided by 2.
Is 0 a prime number?
No. Primes must be greater than 1, and 0 can be divided by every whole number other than 0.
How many prime numbers are there?
Infinitely many. A proof from ancient Greece shows that for any list of primes, you can always find another one not on it.
Is 1 a prime number?
No. It has only one factor, and counting it as prime would spoil the rule that each number has a single prime factorisation.
Which numbers are not prime?
Composite numbers, meaning whole numbers above 1 with more than two factors, plus 0 and 1. Note that 1 is neither prime nor composite.