Prime number/Citable Version

Prime numbers 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 (i.e., one having no fractional or decimal part) that cannot be evenly divided by any numbers but 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. With the exception of $2$, all prime numbers are odd numbers, but not every odd number is prime. For example, $9=3\cdot 3$ and $15=3\cdot 5$, so neither 9 nor 15 is prime. 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.

Definition

A prime numbers is 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, 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 $p\in \mathbb {N}$ 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 $\mathbb {Z}$) is not immediately obvious. In fact, it is a significant result.

Unique factorization

Every integer N > 1 can be written in a unique way as a product of prime factors, up to reordering. to see why this is true, assume that N can be written as a product of prime factors in two ways

$N=p_{1}p_{2}\cdots p_{m}=q_{1}q_{2}\cdots q_{n}$

We may now use a technique known as mathematical induction to show that the two prime decompositions are really the same.

Consider the prime factor $p_{1}$. We know that

$p_{1}\mid q_{1}q_{2}\cdots q_{n}.$

Using the second definition of prime numbers, it follows that $p_{1}$ divides one of the q-factors, say $q_{i}$. Using the first definition, $p_{1}$ is in fact equal to $q_{i}$

Now, if we set $N_{1}=N/p_{1}$, we may write

$N_{1}=p_{2}p_{3}\cdots p_{m}$

and

$N_{1}=q_{1}q_{2}\cdots q_{i-1},q_{i+1}\cdots q_{n}$

In other words, $N_{1}$ is the product of all the $q$'s except $q_{i}$.

Continuing this way, we obtain a sequence of numbers $N=N_{0}>N_{1}>N_{2}>\ldots >N_{n}=1$ where each $N_{\alpha }$ is obtained by dividing $N_{\alpha -1}$ by a prime factor. In particular, we see that $m=n$ and that there is some permutation ("rearrangement") σ of the indices $1,2,\ldots n$ such that $p_{i}=q_{\sigma (i)}$. Said differently, the two factorizations of N must be the same up to a possible rearrangement of terms.

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

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 $p_{i}\not |N$ (read "$P_{i}$ does not divide $N$"; this is 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.

Euler's proof

The Swiss mathematician Leonhard Euler showed how the existence of infinitely many primes could be demonstrated using a rather different approach. The starting point is the fact that the harmonic series

$1+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}+\ldots$

diverges. That is, for any $N>0$, we can choose n such that

$1+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}>N$

A second fact we will need about infinite series is that if $|x|>1$

$1+{\frac {1}{x}}+{\frac {1}{x^{2}}}+\ldots ={\frac {1}{1-x^{-1}}}$

Now, suppose for each prime $p$ we form a series as above, replacing $x$ with $p$, and then take the product of all these series. Multiplying out, each term of the resulting series will have a different combination of prime factors in the denominator, with every possible combination appearing exactly once. Therefore, unique prime factorization gives us

$\prod _{p}{\frac {1}{1-p^{-1}}}=\sum _{n=1}^{\infty }{\frac {1}{n}}$

where the sum on the right is the harmonic series, and the product on the left is extended over all primes p. Now, if there are only finitely many primes, the product on the left has a (finite) value, but the harmonic series (the sum on the right) diverges. This shows that there must be infinitely many primes, after all.

Remark: One might well wonder what point there is in offering multiple proofs of the same result. After all, isn't one enough? In mathematics, the point of writing a proof is not so much to establish that something is true, but to understand why it is true. Euclid's proof is purely algebraic, and ultimately depends on the fact that prime numbers can be characterized in two different ways. Euler's proof, on the other hand, makes use of a couple of facts about infinite series combined with the unique factorization property established above. What is interesting is that these two proofs (and there are many others) use ideas from very different parts of mathematics to arrive at the same result. One has the feeling that this is evidence of particularly deep interconnections between different parts of mathematics.

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 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.

Some unsolved problems

There are many unsolved problems concerning prime numbers. Two 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.

Distribution of prime numbers

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 √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 $\leq x$. This result was demonstrated independently by Jacques Hadamard and Charles de la Vallée Poussin in 1896. An essential part of their proof is the function

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p}{\frac {1}{1-p^{-s}}}.$

Euler introduced this function when considering his proof that there are infinitely many prime numbers. The infinite sums and products in the above definition only converge when s > 1, but a technique called analytic continuation can be used to extend the definition to all values of s, including complex numbers. The function with the extended domain is known as the Riemann zeta function. Hadamard and de la Vallée Poussin proved that this function cannot be zero in certain parts of the complex plane and used this to establish the prime number theorem. A proof not relying on complex analysis proved elusive, even though weaker results on the distribution of prime numbers have long been known. It was only in 1949 that Atle Selberg and Paul Erdös were able to establish the theorem by elementary means.

The location of the values s for which ζ(s) is zero, is one of the biggest mysteries in contemporary mathematics. The Riemann hypothesis states all the zeros of the Riemann zeta function lie on two lines in the complex plane. A proof of the Riemann hypothesis would lead immediately to more precise estimates on how fast the function π(n) grows.

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. ).