The Distribution of the Primes

Basic Definitions

• A natural number is any number in the set {1, 2, 3, 4, 5, … }. Naturals are also called counting numbers.

• A prime number is any natural that is only divisible by 1 or itself (e.g. 7 = 1 × 7 is prime). Any natural that is not prime is called a composite number (e.g. 21 = 3 × 7 is composite).

Set of primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, … }

Note 1: The number 1 is neither prime nor composite. This is a necessary condition for the Fundamental Theorem of Arithmetic (FTA), which states that any natural has a unique prime factorization or, equivalently, that composite numbers can be “broken down” into primes in exactly one way (e.g. 42 = 2 × 3 × 7). If the number 1 were to be considered prime, then this would directly contradict FTA as any composite number could be broken down in 2 distinct ways (e.g. 42 could be factored as 2 × 3 × 7 or as 1 × 2 × 3 × 7).

It follows from this that all naturals greater than 1 are either prime or composite.

Note 2: The oldest and most famous method for obtaining the set of primes is called the Sieve of Erastothenes, named after the Greek mathematician and astronomer Erastothenes (ca. 350 B.C.E.) who served as chief librarian of the famous library in Alexandria and was Euclid’s mentor. [Read more about this method on p. 342.]

• The number “n factorial,” denoted by �!, is the natural defined as follows:

�! = 1 × 2 × 3 × … × (� − 1) × �

For example, 3! = 1 × 2 × 3 = 6 and 5! = 1 × 2 × 3 × 4 × 5 = 120.

Note that (� + 1)! = (� + 1) × �! for any natural �.

• A prime desert of length � is any consecutive list of � naturals that includes only composite numbers (i.e. no primes). For example, the list 8, 9, 10 is a prime desert of length 3 because these 3 naturals are consecutive and none of them is a prime.

An Infinitude of Primes

One of the greatest results in all of mathematics is Euclid’s proof (ca. 300 B.C.) that there are infinitely many primes. Euclid argues this fact by contradiction. First he assumes the very thing he wishes to refute: that there exists a prime larger than any other prime. From that initial assumption he ends up with a contradiction after a sequence of logical steps. As a result, he is forced to conclude that his initial assumption must have been wrong, therefore proving that there cannot be a prime larger than any other prime, or that there must be an infinite number of primes. [Read more about this proof on pp. 342-343]

So we know the list of prime numbers never ends: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

Where Are the Primes?

A natural question to ask is whether we can locate all primes numbers, exactly, on a number line. In other words, is there a formula that yields all the prime numbers? Unfortunately, no such formula exists. Countless mathematicians have cracked their teeth trying to find it... to no avail!

However, the situation is not so dire. A lot is actually known about the distribution of the primes if one is to consider their statistical behavior (i.e. how primes behave collectively and not just individually). This leads to the Prime Number Theorem (proved in 1896 by Jacques Hadamard and Jean de la Vallée-Poussin), a cornerstone of modern-day number theory, and to many other remarkable results.

In the end, a lot is still not known about the distribution of the primes. Perhaps the most important of all mathematical open questions, the Riemann Hypothesis, is intimately related to this topic. A prize of one million dollars from the Clay Institute awaits anyone who is capable of proving the hypothesis.

Note: If you want to see a striking visualization of the distribution of the primes, check out Ulam’s Spiral shown next to the title of this document. This spiral is named after the Polish mathematician Stanislaw Ulam, who came up with it in the 1960’s while he was doodling at a math conference.

Prime Deserts

In the Excursion at the end of Section 6.5 (pp. 343-344), you are exposed to a striking feature of the elusive distribution of the primes. It turns out that you can produce a prime desert of any finite length! Theoretically then, you can eventually find 10 consecutive composites, 1,000 consecutive composites, or even 10 billion consecutive composites, among the set of naturals (which is infinite). The way to achieve this is by using factorials, as presented in the Prime Desert Theorem below.

Prime Desert Theorem (PDT). A prime desert of length � (where � is a natural greater than 2) is given by the following list:

(� + 1)! + 2 (� + 1)! + 3 (� + 1)! + 4 ⋮ (� + 1)! + � (� + 1)! + (� + 1)

So, for example, a prime desert of length 4 is given by the list

(4 + 1)! + 2 = 5! + 2 = 120 + 2 = 122 (4 + 1)! + 3 = 5! + 3 = 120 + 3 = 123 (4 + 1)! + 4 = 5! + 4 = 120 + 4 = 124 (4 + 1)! + 5 = 5! + 5 = 120 + 5 = 125

You can quickly check that none of these 4 naturals is prime.

Note: The Prime Desert Theorem, while remarkable from a theoretical perspective, is not particularly well-suited to finding prime deserts in practice. For example, a simpler prime desert of length 4 is 24, 25, 26, 27.