SafekipediaDirichlet's theorem on arithmetic progressions
Adapted from Wikipedia · Discoverer experience
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, tells us something very interesting about prime numbers. It says that if you take any two positive whole numbers that don't share any factors besides 1 — called coprime numbers — there will always be infinitely many prime numbers in sequences that follow a certain pattern. These sequences are called arithmetic progressions, and they look like this: you start with a number a, and then keep adding another number d over and over again.
For example, if you start with 1 and keep adding 4, you get the sequence 1, 5, 9, 13, 17, and so on. Dirichlet's theorem says that no matter which starting number a you choose (as long as it stays coprime with d), you'll keep finding prime numbers in that list forever. This idea builds on an older discovery by Euclid's theorem, which showed there are infinitely many prime numbers in general.
The theorem was proved in 1837 by the German mathematician Peter Gustav Lejeune Dirichlet. It helps us understand how prime numbers are spread out and shows that they appear in very regular patterns, even when we're looking at special sequences like these.
Examples
Dirichlet's theorem tells us that there are infinitely many prime numbers in certain patterns. For example, numbers like 3, 7, 11, 19, and so on, which follow the pattern 4 times n plus 3, are all prime numbers. These primes keep appearing no matter how far you go in this pattern.
The theorem also helps us understand how these primes behave in sums. If you add the reciprocals of these primes — like 1/3 + 1/7 + 1/11 + 1/19 and so on — the total keeps getting larger and never settles down to a final number. This shows that there are infinitely many primes in this pattern.
| Arithmetic progression | First 10 of infinitely many primes | OEIS sequence |
|---|---|---|
| 2n + 1 | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … | A065091 |
| 4n + 1 | 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, … | A002144 |
| 4n + 3 | 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, … | A002145 |
| 6n + 1 | 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, … | A002476 |
| 6n + 5 | 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, … | A007528 |
| 8n + 1 | 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, … | A007519 |
| 8n + 3 | 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, … | A007520 |
| 8n + 5 | 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, … | A007521 |
| 8n + 7 | 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, … | A007522 |
| 10n + 1 | 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, … | A030430 |
| 10n + 3 | 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, … | A030431 |
| 10n + 7 | 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, … | A030432 |
| 10n + 9 | 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, … | A030433 |
| 12n + 1 | 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, ... | A068228 |
| 12n + 5 | 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, ... | A040117 |
| 12n + 7 | 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, ... | A068229 |
| 12n + 11 | 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, ... | A068231 |
Distribution
See also: Prime number theorem § Prime number theorem for arithmetic progressions
Primes are spread out in a special way across sequences called arithmetic progressions. For a given number d, there are d different progressions, but only those where the starting number a and d share no common factors (other than 1) are important. The number of such progressions is given by Euler's totient function, written as φ(d).
Each of these valid progressions contains an equal share of primes overall. For instance, if d is a prime number q, each of the q−1 progressions (excluding those that are multiples of q) contains about 1/(q−1) of all primes.
History
In 1737, a mathematician named Euler discovered a connection between prime numbers and a special mathematical function. He showed that this function could be expressed using primes in a unique way. Later, in 1775, Euler looked at a specific pattern of numbers and observed something interesting about primes within that pattern.
The full idea behind this theorem was first guessed by another mathematician, Legendre, but it was finally proven by Dirichlet using a new mathematical tool. This work marked the start of a deeper study into the patterns of numbers, called analytic number theory.
Proof
Dirichlet's theorem can be proved by showing that a special value in a mathematical function is not zero. While the full proof uses advanced mathematics, some examples are easier to understand.
One simple example proves that there are infinitely many primes of the form 4_n + 3. Imagine we list all such primes we know. Then we create a new number by multiplying all these primes together and adding 3. This new number will either be a prime itself or have a prime factor that is also of the form 4_n + 3, which means we missed a prime in our list. This shows there must be infinitely many such primes.
Generalizations
Some big ideas in math build on Dirichlet's theorem. The Bunyakovsky conjecture wonders if certain patterns, like x2 + 1, can make infinitely many prime numbers — but we still don’t know the answer. Other ideas, like Dickson's conjecture and Schinzel's hypothesis H, look at many patterns together.
In more advanced math, Dirichlet's ideas also fit into bigger theories. Linnik’s theorem tells us about how big the first prime in a pattern might be. There are even versions of Dirichlet's work that apply to special kinds of math patterns called dynamical systems.
This article is a child-friendly adaptation of the Wikipedia article on Dirichlet's theorem on arithmetic progressions, available under CC BY-SA 4.0.