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.

	2	3	5	7	11	13	17	19	23
29	31	37	41	43	47	53	59	61	67
71	73	79	83	89	97	101	103	107	109
113	127	131	137	139	149	151	157	163	167
173	179	181	191	193	197	199	211	223	227
229	233	239	241	251	257	263	269	271	277
281	283	293	307	311	313	317	331	337	347
349	353	359	367	373	379	383	389	397	401
409	419	421	431	433	439	443	449	457	461
463	467	479	487	491	499	503	509	521	523
541	547	557	563	569	571	577	587	593	599
601	607	613	617	619	631	641	643	647	653
659	661	673	677	683	691	701	709	719	727
733	739	743	751	757	761	769	773	787	797
809	811	821	823	827	829	839	853	857	859
863	877	881	883	887	907	911	919	929	937
941	947	953	967	971	977	983	991	997

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.

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