Prime Numbers: List, How To Check Them And Fun Facts

What Is A Prime Number?

A prime number has exactly two factors: 1 and the number itself. A factor is a whole number that divides into another with no remainder.

Compare 7 and 8:

Another way to picture it: try to arrange a prime number of objects into a rectangle. Seven coins can only sit in one row of 7. Eight coins can form 2 rows of 4. Primes refuse to make any rectangle except a single line.

Why 1 Is Not A Prime Number

The number 1 is not prime because it has only one factor: itself. A prime needs exactly two different factors.

There is also a deeper reason. Every whole number above 1 can be broken into primes in only one way, such as 12 = 2 × 2 × 3. If 1 counted as prime, you could add as many 1s as you like (12 = 1 × 1 × 2 × 2 × 3), and that one-way rule would fall apart. Leaving 1 out keeps the math clean.

Why 2 Is The Only Even Prime

The number 2 is the smallest prime and the only even one. Every other even number can be divided by 2, which gives it at least three factors: 1, 2 and itself.

So 4, 6, 8, 10 and every even number after them are composite. That means all primes after 2 are odd. But not all odd numbers are prime: 9, 15, 21 and 25 are odd and composite.

Prime Vs Composite Numbers

Every whole number greater than 1 is either prime or composite. A composite number has more than two factors, so it can be made by multiplying two smaller whole numbers.

The numbers 0 and 1 are neither prime nor composite. Zero has endless factors, since every number divides into 0, and 1 has just one factor.

List Of Prime Numbers From 1 To 100

There are exactly 25 prime numbers between 1 and 100. Here they are, grouped by tens:

The primes between 100 and 200 are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197 and 199. That is 21 more.

Notice how the gaps start to grow. The 1 to 10 row has four primes, while 91 to 100 has only one.

How To Check If A Number Is Prime

To check if a number is prime, try dividing it by each prime number up to its square root. If none of them divide evenly, the number is prime. This is the same method a basic prime number checker uses behind the scenes.

Step 1: Use Quick Divisibility Rules

Start by ruling out the easy cases. These rules let you test small primes without long division.

If any rule fits, the number is composite and you can stop. For example, 51 looks prime to many people, but 5 + 1 = 6, so 51 = 3 × 17.

Step 2: Test Primes Only Up To The Square Root

You never need to test past the square root of the number. If a number has a factor bigger than its square root, it must also have one smaller than the square root, and you would already have found that one.

Example: Is 221 prime?

1. √221 is about 14.9, so test primes up to 13: 2, 3, 5, 7, 11, 13.
2. 2, 3, 5, 7 and 11 do not divide evenly.
3. 221 ÷ 13 = 17 exactly.
4. 221 is composite (13 × 17).

Example: Is 1,009 prime?

1. √1,009 is about 31.8, so test primes up to 31.
2. That is 11 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
3. None divide evenly.
4. 1,009 is prime.

A square root calculator tells you where to stop testing when the number is large.

Worked Examples: Prime Or Not?

These are the numbers people get wrong most often. Each one looks prime but is not, except where noted.

Numbers like 91, 143 and 899 trick people because their smallest factor is fairly large. The square root method catches them every time.

How Computers Test Very Large Numbers

Checking every prime up to the square root works for small numbers, but it is far too slow for numbers with hundreds or millions of digits. Computers use smarter tests instead.

Probable prime tests. A test based on Fermat's little theorem picks a number a and checks whether aⁿ⁻¹ leaves a remainder of 1 when divided by n. Every prime passes. Most composite numbers fail quickly, so a pass means "almost certainly prime."

Watch out for fakes. A few composite numbers, called Carmichael numbers, pass the basic Fermat test for every base a that shares no factor with them. The smallest is 561 = 3 × 11 × 17. Stronger tests, such as the Miller-Rabin test, close this gap and are what most software uses today.

Special tests for special primes. Mersenne numbers (2ⁿ – 1) have their own fast test, the Lucas-Lehmer test. That is one reason the largest known primes are almost always Mersenne primes.

The Sieve Of Eratosthenes

The Sieve of Eratosthenes is a simple way to find every prime up to a chosen number by crossing out multiples. It is named after Eratosthenes, a Greek scholar from the 3rd century BCE.

Here is how to find all primes up to 100:

1. Write the numbers 2 to 100 in a grid.
2. Circle 2, then cross out every multiple of 2 (4, 6, 8 and so on).
3. Circle the next number not crossed out, which is 3. Cross out its multiples (6, 9, 12 and so on).
4. Circle 5 and cross out its multiples. Then do the same with 7.
5. Stop there. Since √100 = 10, any composite number up to 100 has a factor of 7 or less.
6. Every number left uncrossed is prime.

You will end up with the same 25 primes from the list above. The sieve is great for classrooms because students see the pattern of multiples forming stripes and columns on the grid.

Prime Factorization: Breaking Numbers Into Primes

Prime factorization means writing a number as a product of primes. Every whole number above 1 has exactly one prime factorization, a rule known as the Fundamental Theorem of Arithmetic.

Factor tree for 360:

1. 360 = 36 × 10
2. 36 = 6 × 6 = 2 × 3 × 2 × 3
3. 10 = 2 × 5
4. Collect the primes: 2 × 2 × 2 × 3 × 3 × 5
5. Write with exponents: 360 = 2³ × 3² × 5

You can start the tree with any pair of factors, like 360 = 4 × 90 or 360 = 12 × 30. The final list of primes will always be the same.

Prime factorization is the engine behind several everyday math skills:

• Finding the greatest common factor, which you can check with a GCF calculator
• Finding the least common multiple, which an LCM calculator works out in seconds
• Reducing fractions to lowest terms, as the fraction simplifier does
• Listing every factor of a number, which a factor calculator handles for large values

Prime Vs Coprime: Not The Same Thing

Two numbers are coprime (also called relatively prime) when their only common factor is 1. The numbers themselves do not need to be prime.

Coprime pairs matter when simplifying fractions. A fraction is in lowest terms exactly when its numerator and denominator are coprime, so 8/15 cannot be reduced even though neither number is prime.

Types Of Prime Numbers

Mathematicians have named many special groups of primes based on patterns they follow. These are the ones you are most likely to meet.

There are 8 twin prime pairs below 100: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) and (71, 73).

Mersenne primes deserve a closer look. 2ⁿ – 1 can only be prime when n is itself prime, but a prime n does not guarantee it. For example, 2¹¹ – 1 = 2,047 = 23 × 89. That is why each new Mersenne prime takes a huge amount of computer time to confirm.

Patterns In Prime Numbers

Primes look random at first, but they follow a few reliable patterns that help you spot them faster.

Last digits. After 2 and 5, every prime ends in 1, 3, 7 or 9. A number ending in 0, 2, 4, 6 or 8 is even, and one ending in 5 is divisible by 5.

The 6k ± 1 rule. Every prime greater than 3 is one more or one less than a multiple of 6. For example, 29 = 30 – 1 and 37 = 36 + 1. The reverse is not true, though: 25 = 24 + 1 fits the pattern but is 5 × 5.

Primes thin out. They get rarer as numbers grow, but they never run out.

Prime Numbers Facts That Surprise Most People

Prime numbers have a long history and a surprising number of real-world jobs. These prime numbers facts are some of the most interesting.

1. The largest known prime has over 41 million digits. It is 2¹³⁶²⁷⁹⁸⁴¹ – 1, found on October 12, 2024, by Luke Durant, a researcher from San Jose, California. It has 41,024,320 digits.
2. Volunteers find most record primes. That record came from the Great Internet Mersenne Prime Search (GIMPS), a project where people donate computer time to test huge numbers.
3. Only 52 Mersenne primes are known. Every new one is a major event, because testing a single candidate can take days of computing.
4. Your online security depends on primes. Encryption systems such as RSA multiply two very large primes together. Multiplying is fast, but working backward to find the two primes from their product is extremely slow, which keeps messages private.
5. Cicadas may use primes to survive. Periodical cicadas in eastern North America come out of the ground every 13 or 17 years. One theory is that these prime cycles make it harder for predators with shorter cycles to line up with them.
6. 2 and 3 are the only primes that sit next to each other. Any two numbers in a row include an even number, and 2 is the only even prime. So no other pair of neighbors can both be prime.
7. The number 2,027 is prime. So next year is a prime year. The last prime year was 2017.
8. Prime gaps can be as long as you like. For any length you pick, there is a stretch of that many composite numbers in a row somewhere on the number line.
9. The digits of a prime never add up to a multiple of 3, except for 3 itself. If they did, the number would be divisible by 3.

Famous Unsolved Prime Number Problems

Some of the hardest open questions in math are about primes. They are easy to state but no one has proved them.

Computers have checked Goldbach's conjecture for a huge range of even numbers, and it has always held. But checking examples is not a proof, which is why it remains open.

When Students Learn About Prime Numbers

In schools that follow the Common Core State Standards, students start working with prime numbers in grade 4. They build on the idea in grade 6 and beyond.

A fun way to practice at home is a "prime hunt." Pick a page of a book, read every page number, and say "prime" or "composite" out loud. Kids get quick feedback and learn the small primes by heart.

Common Mistakes To Avoid

Most mistakes with prime numbers come from assuming odd numbers are prime or forgetting the special cases of 0, 1 and 2.

Practice Problems With Answers

Try these, then check below.

1. Is 47 prime?
2. Is 87 prime?
3. List the primes between 50 and 70.
4. Find the prime factorization of 84.
5. Is 323 prime? (Hint: √323 is about 18)
6. What is the next prime after 113?

Answers

1. Yes. √47 is about 6.9, and 2, 3 and 5 do not divide it.
2. No. 8 + 7 = 15, which is divisible by 3, so 87 = 3 × 29.
3. 53, 59, 61 and 67.
4. 84 = 2 × 2 × 3 × 7, or 2² × 3 × 7.
5. No. 323 = 17 × 19.

1. Every number from 114 to 126 is composite.

Summary

A prime number is a whole number greater than 1 with exactly two factors, 1 and itself. 2 is the smallest and the only even prime, and 1 is not prime. There are 25 primes up to 100 and infinitely many in total.

To check a number, use divisibility rules first, then test primes up to the square root. Prime factorization breaks any number into primes in exactly one way, which powers GCF, LCM and fraction work.

Next step: use the Sieve of Eratosthenes to find every prime up to 200 on your own, then compare your list with the one above.

Frequently Asked Questions