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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 , all prime numbers are odd numbers, but not every odd number is prime. For example,  and , 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  is prime if for any  such that  (read p divides ab), either  or .

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 ) is not immediately obvious. In fact, it is a significant result.[1]

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



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 . We know that



Using the second definition of prime numbers, it follows that  divides one of the q-factors, say . Using the first definition,  is in fact equal to 

Now, if we set , we may write



and



In other words,  is the product of all the 's except .

Continuing this way, we obtain a sequence of numbers  where each  is obtained by dividing  by a prime factor. In particular, we see that  and that there is some permutation ("rearrangement") σ of the indices  such that . 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. Another one, due to Leonhard Euler, is described in another section of the article.

Euclid's proof is a proof by contradiction. Suppose the set of prime numbers is finite, say , and let



then for each  we know that  (read " does not divide "; 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.

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, ...,  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  numbers to be sure. This algorithm can be improved by realizing the following: if n has a divisor a that is larger than , there must be another divisor b that is smaller than . Thus, it is sufficient to look for a divisor up to . 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 :
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  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. For example, if you choose the even number 48, you can find  where 41 and 7 are prime numbers.

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



where  is the number of primes . 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



This function was already considered by the 18th century Swiss mathematician Leonhard Euler, who realized that



where the product on the right goes over all prime numbers:



To see why this is so, we use the formula for the sum of a geometric series to write the product as



Multiplying out this product, we get a sum of terms of the form



where , …  are prime numbers and , …  are positive integers. Euler's formula equating the sum and the product now follows from the fact that every number has a unique prime factorization. This also indicates the connection between the function ζ and the prime numbers.

Euler used his formula to give an alternative proof that there are infinitely many primes. If we take , we get



The sum on the left is the harmonic series, which diverges. If the number of primes were finite, then the product would evaluate to a finite number. This contradiction shows that there are infinitely many primes.

The above definition for the function ζ only works when s > 1, because the infinite sum diverges otherwise, 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[2][3]. 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.

References and Notes

1. The Euclidean algorithm may be used to show that  is a principal ideal domain, and this implies that irreducibles are prime.
2. Ribenboim, Introduction to Analytic Number theory
3. Scharlau and Opolka, From Fermat to Minkowski