Prime number/Citable Version

The prime number 11 illustrated with square tiles: 12 squares can be arranged into a rectangle with sides of length 3 and 4, so 12 is not a prime number. There is no way to form a full rectangle more than one square wide with 11 squares, so 11 is a prime number.

A prime number is a whole number that can be evenly divided by exactly two numbers, namely 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. Another characterization is that a prime number $p$ cannot be factored as the product of two numbers $p=a\times b$, except for the "boring" factorizations $p=1\times p=p\times 1$ that all numbers possess. With the exception of 2, all prime numbers are odd numbers, but not every odd number is prime. For example, $9=3\times 3$ and $15=3\times 5$, so neither 9 nor 15 is prime. By the strict mathematical definition, the number 1 is not considered to be prime (although this is a matter of definition, and mathematicians in the past often did consider 1 to be a prime).

Every integer $N>1$ can be written in a unique way, up to reordering, as a product of prime factors. For example, we can write $5040=7\times 5\times 3\times 3\times 2\times 2\times 2\times 2$, and all of the factors in the product are prime numbers; moreover, there is no other way of doing this other than simply rearranging the prime factors we already have, such as $5040=2\times 3\times 2\times 7\times 2\times 3\times 2\times 5$. Because of this, prime numbers play a role in arithmetic analogous to that of atoms in chemistry.

The study of prime numbers has a long history, going back to ancient times, and it remains an active part of number theory (a branch of mathematics) today. It is commonly believed that the study of prime numbers is an interesting, but not terribly useful, area of mathematical research. While this used to be the case, the theory of prime numbers has important applications now. Understanding properties of prime numbers and their generalizations is essential to modern cryptography, and to public key ciphers that are crucial to Internet commerce, wireless networks and, of course, military applications. Less well known is that other computer algorithms also depend on properties of prime numbers.

There are infinitely many primes

One basic fact about the prime numbers is that there are infinitely many of them. In other words, the list of prime numbers 2, 3, 5, 7, 11, 13, 17, ... doesn't ever stop. There are a number of ways of showing that this is so, but one of the oldest and most familiar proofs goes back go Euclid.

Euclid's proof is a proof by contradiction. Suppose the set of prime numbers is finite, say $\{p_{1},p_{2},p_{3},\ldots ,p_{n}\}$, and let

$N=p_{1}p_{2}\cdots p_{n}+1$

then for each $i\in 1,\ldots ,n$ we know that $p_{i}$ does not divide $N$, because the remainder is 1. This means that N is not divisible by any prime, which is impossible. This contradiction shows that our assumption that there must only be a finite number of primes must have been wrong and thus proves the theorem.

Locating primes

How can we tell which numbers are prime and which are not? It is sometimes possible to tell that a number is not prime from looking at its digits: for example, any number larger than 2 whose last digit is even is divisble by 2 and hence not prime, and any number ending with 5 or 0 is divisible by 5. Therefore, any prime number larger than 5 must end with 1, 3, 7 or 9. This check can be used to rule out the possibility of a randomly chosen number being prime roughly half of the time, but a number that ends with 1, 3, 7 or 9 could have a divisor that is harder to spot.

To find large prime numbers, we must use a systematic procedure — an algorithm. Nowadays, prime-finding calculations are performed by computers, often using very complicated algorithms, but there are simple algorithms that can be carried out by hand if the numbers are small. In fact, the simplest methods for locating prime numbers are some of the oldest algorithms, known since antiquity. Two classical algorithms are called trial division and the sieve of Eratosthenes.

Trial division

Trial division consists of systematically searching the list of numbers 2, 3, ..., $n-1$ for a divisor; if none is found, the number is prime. If n has a small divisor, we can quit as soon as we've found it, but in the worst case — if n is prime — we have to test all $n-2$ numbers to be sure. This algorithm can be improved by realizing the following: if n has a divisor a that is larger than ${\sqrt {n}}$, there must be another divisor b that is smaller than ${\sqrt {n}}$. Thus, it is sufficient to look for a divisor up to ${\sqrt {n}}$. This makes a significant difference: for example, we only need to try dividing by 2, 3, ..., 31 to verify that 997 is prime, rather than all the numbers 2, 3, ..., 996. Trial division might be described as follows using pseudocode:

Algorithm: trial division

Given n,

For each i = 2, 3, ... less than or equal to ${\sqrt {n}}$:

If i divides n:

Return "n is not prime"

Else:

Continue with the next i

When all i have been checked:

Return "n is prime"

Sieve of Eratosthenes

The Sieve of Eratosthenes not only provides a method for testing a number to see if it is prime, but also for enumerating the (infinite) set of prime numbers. The idea of the method to write down a list of numbers starting from 2 ranging up to some limit, say

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

The first number (2) is prime, so we mark it, and cross out all of its multiples

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

The smallest unmarked number is 3, so we mark it and cross out all its multiples (some of which may already have been crossed out)

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

The smallest unmarked number (5) is the next prime, so we mark it and cross out all of its multiples

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Notice that there are no multiples of 5 $(\leq 20)$ that haven't already been crossed out, but that doesn't matter at this stage. Proceeding as before, we add 7, 11, 13, 17 and 19 to our list of primes

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

We have now found all prime numbers up to 20.

Distribution of prime numbers

It is evident that the prime numbers are randomly distributed but, unfortunately, we don't know what 'random' means. - R. C. Vaughan

The list of prime numbers suggests that they thin out the further you go: 44% of the one-digit numbers are prime, but only 23% of the two-digit numbers and 16% of the three-digit numbers. The trial division method explained above provides an intuitive explanation. To test whether a number $n$ is prime, you have to try whether it can be divided by all numbers between 2 and ${\sqrt {n}}$. Large numbers have to undergo more tests, so fewer of them will be prime.

The Prime Number Theorem explains how fast the prime numbers thin out. It says that if you are looking around the number $n$, about one in every $\log n$ numbers is prime (here, $\log n$ denotes the natural logarithm of $n$). The formal statement of the prime number theorem is

$\lim _{x\to \infty }{\frac {\pi (x)\log x}{x}}=1$

where $\pi (x)$ is the number of primes $\leq x$.

Some unsolved problems

There are many unsolved problems concerning prime numbers. A few such problems (posed as conjectures) are:

The twin prime conjecture

Twin primes are pairs of prime numbers differing by 2. Examples of twin primes include 5 and 7, 11 and 13, and 41 and 43. The Twin Prime Conjecture states that there are an infinite number of these pairs. It remains unproven.

The Goldbach conjecture

The Goldbach conjecture is that every even number greater than 2 can be expressed as the sum of two primes. For example, if you choose the even number 48, you can find $48=41+7$ where 41 and 7 are prime numbers.

Primes of special forms

• It is not known whether there are infinitely many primes of the form $2^{n}-1$ (called Mersenne primes). These primes arise in the study of perfect numbers, and factors of numbers of the form $2^{n}-1$ (sometimes called Mersenne numbers) are a fruitful source of large prime numbers.
• It is not known whether there are infinitely many primes of the form $2^{n}+1$ (called Fermat primes). Fermat primes arise in elementary geometry because if $p$ is a Fermat prime, it is possible to construct a regular $p$-gon with a ruler and compass. In particular, it is possible to construct a regular 17 sided polygon (or 17-gon, for short) with a ruler and compass.
• It is not known whether there are infinitely many primes of the form $n^{2}+1$.

Alternative definition

A prime number is usually defined as a positive integer other than 1 that is (evenly) divisible only by 1 and itself.

There is another way of defining prime numbers, and that is that a number is prime if whenever it divides the product of two numbers, it must divide one of those numbers. A nonexample (if you will) is that 4 divides 12 (i.e. is a factor of 12), but 4 does not divide 2 and 4 does not divide 6 even though 12 is 2 times 6. This means that 4 is not a prime number. We may express this second possible definition in mathematical notation as follows: A number $p\in \mathbb {N}$ (natural number) is prime if for any $a,b\in \mathbb {N}$ such that $p|ab$ (read p divides ab), either $p|a$ or $p|b$.

If the first characterization of prime numbers is taken as the definition, the second is derived from it as a theorem, and vice versa. The equivalence of these two definitions (in the integers $\mathbb {Z}$) is not immediately obvious. In fact, it is a significant result.

Further reading

• Apostol, Tom M. (1976). Introduction to Analytic Number Theory. Springer-Verlag. ISBN 0-387-90163-9. 
• Ribenboim, Paulo (2004). The Little Book of Bigger Primes, second edition. Springer-Verlag. ISBN 0-387-20169-6. 
• Scharlau, Winfried; Hans Opolka (1985). From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development. Springer-Verlag. ISBN 0-387-90942-7. 

(Note that Scharlau and Opolka was originally published as: Scharlau, Winfried; Hans Opolka (1980). Von Fermat bis Minkowski: Eine Vorlesung über Zahlentheorie und ihre Entwicklung. Springer-Verlag. ).