SafekipediaDirichlet's theorem on arithmetic progressions
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In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, tells us something interesting about prime numbers. It says that if you take any two whole numbers that don't share any factors besides 1 — called coprime numbers — there will always be infinitely many prime numbers in special patterns called arithmetic progressions.
These sequences 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.
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 in sequences called arithmetic progressions. For a given number d, there are d different progressions. Only the progressions that start with a number a that does not share any factors (other than 1) with d are important. The number of these special progressions is found using Euler's totient function, written as φ(d).
Each of these special progressions has about the same number of primes. For example, if d is a prime number q, each of the q−1 progressions (not including those that are multiples of q) contains about 1/(q−1) of all primes.
History
In 1737, a mathematician named Euler found a link between prime numbers and a special math function. He showed that this function could be written using primes in a unique way.
Later, in 1775, Euler studied a specific pattern of numbers and noticed something interesting about the primes in that pattern.
The full idea behind this theorem was first guessed by another mathematician, Legendre, but it was proven by Dirichlet using a new math tool. This work started 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 grow from Dirichlet's theorem. The Bunyakovsky conjecture asks 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.