Prime Numbers Chart

Why 1 is not prime, and why 2 is the only even prime

A prime needs two different divisors, and 1 has only one. There is also a deeper reason: if 1 counted as prime, 6 could be written as 2 × 3, 1 × 2 × 3, 1 × 1 × 2 × 3 and so on, and the fundamental theorem of arithmetic (every whole number above 1 has exactly one prime factorization) would be false. Every other even number is divisible by 2, so it has at least three divisors; that makes 2 the only even prime. Zero is not prime either, since every number divides it.

How to tell if a number is prime

• Trial division is the standard test: divide by each prime up to the square root of the number. If none divides it evenly, it is prime. The square root is enough because a composite number n = a × b has one factor at most √n. For 91 the square root is about 9.5, so only 2, 3, 5 and 7 need checking, and 7 divides it: 91 = 7 × 13. That is why 91 looks prime but is not.
• The Prime checker tab runs this test for any whole number up to 1,000,000,000,000 and shows the factorization. For example 360 = 2^3 × 3^2 × 5, 600851475143 has 4 prime factors (71 × 839 × 1471 × 6857) and 999999000001 is prime.

The sieve of Eratosthenes

The sieve finds every prime up to a limit without any division. Write the numbers from 2 up, keep the first uncrossed number, cross out its multiples starting from its square, then move to the next uncrossed number. For 1 to 100 you stop after 7, because the next prime, 11, has a square above 100 and every composite up to 100 has a prime factor of at most 10. The Sieve tab animates each step, and you can pause it or step with Next. If your device asks for reduced motion, autoplay is off and you step by hand.

Patterns in the primes

Twin primes are primes two apart, such as 11 and 13. There are 8 twin prime pairs up to 100 and 35 up to 1000. Primes can also leave long gaps: the widest gap below 1000 is 20, between 887 and 907. Apart from 2 and 3, every prime is one more or one less than a multiple of 6, and every prime above 5 ends in 1, 3, 7 or 9. Among the 168 primes to 1000 the last digits split as 40 ending in 1, 42 in 3, 46 in 7 and 38 in 9.

Prime factorization and the fundamental theorem

Every whole number above 1 is either prime or a product of primes in exactly one way, apart from the order. Factor trees and repeated division both find it. Writing repeated primes with exponents gives the short form, so 360 is 2^3 × 3^2 × 5 and has 24 divisors. Prime factorizations are how you find a greatest common factor or a least common multiple, and they underlie the encryption that protects online payments. The factorization sheet in the Printables tab lists every number from 2 to 100.