The Distribution of the Primes
Basic Definitions
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A natural number is any number in the set {1, 2, 3, 4, 5, ...}. Naturals are also called
counting numbers.
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A prime number is any natural that is only divisible by 1 or itself (e.g. 7 = 7 x 1 is prime).
Any natural that is not prime is called a composite number (e.g. 15 = 5 x 3 is composite).
The set of primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...}
Note: The number 1 is considered to be neither prime nor composite. This is a
necessary condition for the Fundamental Theorem of Arithmetic. This very important theorem
states that any natural has a unique prime factorization (e.g. 42 = 2 x 3 x 7). In other
words, there is only one way to “break down” a composite number into primes.
It follows from this that all naturals greater than 1 are either prime or composite.
•
The number “n factorial,” denoted by n!, is the natural defined as follows:
n! = n x (n - 1) x (n – 2) x ... x 3 x 2 x 1
For example,
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3! = 3 x 2 x 1 = 6
and
5! = 5 x 4 x 3 x 2 x 1 = 120.
A prime desert of length n is any consecutive list of n 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 (or, alternatively, all
the composites)? 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 $1,000,000.00 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 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. A prime desert of length n (where n is a natural greater than 2) is
given by the following list:
(n + 1)! + 2
(n + 1)! + 3
(n + 1)! + 4
:
(n + 1)! + n
(n + 1)! + (n + 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.