# What Is a Prime Number? How to Tell If a Number Is Prime A prime number is divisible only by itself and 1. There are 25 prime number less than 100.

A prime number is a natural number that can only be divided, without a remainder, by itself and 1. In other words, a prime number has exactly two factors. For example, 13 is only divisible by 13 and 1. In contrast, a composite number is a natural number that can be divided evenly by any number besides itself and 1. A composite number has more than two factors. For example, 14 is divisible by 1, 2, 7, and 14.

Here is a list of the prime numbers up to 1000 and a look at how to tell if a number is prime.

### Interesting Prime Number Facts

• The state of being prime is called primality.
• There are an infinite number of prime numbers.
• Zero and one are not prime numbers.
• Two is the only even prime number.
• Two and three are the only consecutive prime numbers.
• No prime number greater than five ends in 5.
• No prime number ends with 0.
• Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two prime numbers.
• Every prime number greater than 2 and 3 can be represented as 6n+1 or 6n-1.
• Prime Number Theorem: The probability that a number is prime is inversely proportional to its number of digits.
• Lemoine’s Conjecture: Any odd integer greater than 5 can be expressed as the sum of an off prime and an even semiprime. A semiprime is the product of two prime numbers.

### Prime Numbers Up to 1000

The smallest prime number is 2, which is also the only even prime number. Here is a table of all the prime numbers up to 1000.

### Is 1 a Prime Number?

The number 1 is not usually considered a prime number. It’s also not a composite number.

• 1 is not a prime number because it does not have exactly two positive factors.
• 1 is not a composite number because it does not have more than two factors.

Note: There are some people who argue 1 is a prime number because it’s divisible by itself and 1 (even though these two values are the same thing).

### How to Tell If a Number Is Prime

There are a few different ways to tell whether or not a number is prime. The methods are called primality tests, even though some of them actually test whether a number is composite.

Basically, you test whether a number n is evenly divisible by any prime number between 2 and √n. This is called trial division or factorization.

• No prime number ends with 0.
• No even number except 2 is prime. If a number ends with 0, 2, 4, 6, or 8, it’s a composite number.
• If the sum of the digits of a number are divisible by 3, it’s a composite number. A prime number can end with 3.
• No prime number ends with 5, except 5.
• If a number passes all of these tests, check to see if it’s divisible by prime numbers smaller than it. It’s not necessary to check prime numbers greater than n. Start with 3, 5, 7, 11, and work your way up to n.
• Check whether or not a number can be expressed as either 6n+1 or 6n-1. For example, the prime number 11 can be written as 6(2)-1.

### Examples: Finding a Prime Number Using Factorization

#### Example 1:

• Is 15874 prime?
• Right away, you can see it’s not prime because it ends with an even number.

#### Example 2:

• Is 26577 a prime number?
• It does not end in 0, 2, 4, 6, 8.
• The sum of the digits 2 + 6 + 5 + 7 + 7 = 27.
• 27 is divisible by 3, so 26577 is not prime.

#### Example 3:

• Is 103 a prime number?
• It does not end in 0, 2, 4, 6, 8.
• It does not end in 5.
• The sum of the digits 1 + 0 + 3 = 4. It is not divisible by 3.
• The 103 is ~10.14. So, check to see if 103 is divisible by other primes under 10.
• 103 is not evenly divisible by 7.
• 103 is a prime number!

### What Is the Largest Prime Number?

There are an infinite number of prime numbers, so computers discover new primes (slowly, because it takes a lot of computing power). To date, the largest prime number is 282,589,933-1. The Great Internet Mersenne Prime Search (GIMPS) found this prime on December 7, 2018.

• Adler, Irving (1960). The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space. Golden Press.
• Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (2nd ed.). Springer. ISBN 0-387-25282-7.
• Dudley, Underwood (1978). “Section 2: Unique factorization“. Elementary Number Theory (2nd ed.). W.H. Freeman and Co. ISBN 978-0-7167-0076-0.
• GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1“. Mersenne Research, Inc.
• Ziegler, Günter M. (2004). “The great prime number record races”. Notices of the American Mathematical Society. 51 (4): 414–416.