What Is a Prime Number? How to Tell If a Number Is Prime

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 2^82,589,933-1. The Great Internet Mersenne Prime Search (GIMPS) found this prime on December 7, 2018.

References

• 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: 2^82,589,933-1“. Mersenne Research, Inc.
• Ziegler, Günter M. (2004). “The great prime number record races”. Notices of the American Mathematical Society. 51 (4): 414–416.