*Originally submitted on November 2014, by Andrea Casanova for the ACIT Journal*

*Rewritten into article form below by*

**Paschal Alaemezie****Introduction**

Imagine you are a treasure hunter, looking for rare and precious gems hidden in a vast desert. You have a map that tells you the location of some of these gems, but not all of them. You know that the gems are arranged in a pattern, but you don’t know what the pattern is. How can you find more gems without wandering in the desert?

This is similar to the problem that mathematicians face when they study prime numbers, those numbers that can only be divided by themselves and one, such as 2, 3, 5, 7, 11, and so on. Prime numbers are the building blocks of all other numbers, and they have many fascinating and mysterious properties. One of the most intriguing questions about prime numbers is how they are distributed among the natural numbers. Are there any patterns or regularities that we can use to predict where the next prime number will be?

One of the most remarkable discoveries in this area is Dirichlet’s theorem on arithmetic progressions, which states that for any two positive coprime integers *a* and *d*, there are infinitely many primes of the form *a + nd*, where *n* is also a positive integer. (“Dirichlet’s theorem on arithmetic progressions – Wikipedia”) In other words, there are infinitely many primes that are congruent to a modulo* d*. The numbers of the form *a + nd* form an arithmetic progression, which is a sequence of numbers with a constant difference between consecutive terms. For example, if we take *a* = 1 and* d* = 4, we get the arithmetic progression 1, 5, 9, 13, 17, 21, …, and Dirichlet’s theorem tells us that there are infinitely many primes in this sequence, such as 5, 13, 17, ….

Dirichlet’s theorem is a remarkable generalization of Euclid’s theorem that there are infinitely many prime numbers. It also gives us a way to find more prime numbers by looking at certain arithmetic progressions. But how did Dirichlet come up with this theorem? What tools and techniques did he use to prove it? And what are the implications and applications of this theorem for mathematics and beyond? These are some of the questions that we will explore in this article.

In this article, we will:

- Focus on the main ideas and insights behind Dirichlet’s theorem, rather than on the technical details and proofs.
- Learn about the history and the proof of this remarkable theorem, as well as its applications and implications for modern mathematics.
- You will also see how this theorem connects with other branches of mathematics, such as complex analysis, algebra, and cryptography.

Are you ready to embark on this journey from Euclid to modern number theory? Let’s get started!

**Prerequisite**

To fully appreciate and understand Dirichlet’s theorem, you will need some background knowledge of number theory, which is the branch of mathematics that studies the properties and relationships of integers. You will also need some familiarity with modular arithmetic, which is a way of doing arithmetic with remainders after division. For example, if we divide 7 by 3, we get a quotient of 2 and a remainder of 1. We can write this as 7 ≡ 1 (mod 3), which means that 7 and 1 have the same remainder when divided by 3. Modular arithmetic allows us to work with congruence classes modulo *d*, which are sets of numbers that have the same remainder when divided by *d*. For example, the congruence class of 1 modulo 4 is {…, −7, −3, 1, 5, 9, …}.

If you are not familiar with these concepts, don’t worry. We will provide some examples and explanations along the way to help you follow along. But if you want to dive deeper into the details and proofs of Dirichlet’s theorem, you might want to check out some of the resources made available at the end of this article.

**What is Dirichlet’s theorem on arithmetic progressions?**

Dirichlet’s theorem on arithmetic progressions is a statement about the distribution of prime numbers in arithmetic progressions. It was first proved by Peter Gustav Lejeune Dirichlet (1805-1859), a German mathematician who made significant contributions to number theory, analysis, and algebra. Dirichlet was inspired by the work of his predecessors, such as Carl Friedrich Gauss (1777-1855), who conjectured the theorem without providing a proof. He was particularly interested in the distribution of prime numbers and their relation to other number-theoretic functions. He was also fascinated by the sieve of Eratosthenes and its patterns.

In his paper “On the use of Fermat’s theorem in number theory,” published in 1837, Dirichlet stated and proved his famous theorem on arithmetic progressions. He used a combination of algebraic and analytic methods, involving modular arithmetic, quadratic reciprocity, Dirichlet characters – special functions that encode information about congruences, – and complex analysis. His proof was very ingenious, elegant, and based on a novel technique that involved complex analysis and Dirichlet characters.

However, this proof is very technical and difficult to follow for non-experts. We will not go into the details of his proof here, but we will try to explain some of the main ideas and insights behind it.

The theorem can be stated as follows:

**Dirichlet’s theorem on arithmetic progressions:** Let* a* and *N* be positive integers such that (*a, N*) =1, where (*a, N*) denotes the greatest common divisor of* a* and *N*. Then there exist infinitely many primes p of the form *p = a + Nk*, for some integer* k*. In other words, there are infinitely many primes* p* such that *p ≡ a* (mod *N*).

For example, if we take *a* = 3 and *N* = 4, then (*a, N*) = 1, and we can find infinitely many primes *p* such that *p ≡ *3 (mod 4). Some of these primes are 3, 7, 11, 19, 23, 31, … If we take* a* = 2 and *N* = 4, then (*a, N*) = 2, and we cannot find any prime *p* such that *p ≡* 2 (mod 4). The only prime that is divisible by 4 is 2 itself.

The theorem tells us that every arithmetic progression with relatively prime terms contains infinitely many primes.

**Why is Dirichlet’s theorem on arithmetic progressions important and relevant?**

Dirichlet’s theorem on arithmetic progressions is one of the most important and influential results in number theory, as it reveals a deep and surprising connection between prime numbers and congruences. Prime numbers are the building blocks of all numbers, as every positive integer can be uniquely written as a product of primes. Congruences are a way of comparing numbers by their remainders when divided by a fixed modulus. For example, 15 ≡ 3 (mod 4) means that 15 and 3 have the same remainder when divided by 4, which is 3. Congruences are useful for simplifying calculations and solving equations involving integers.

Dirichlet’s theorem shows that prime numbers are not randomly distributed among the natural numbers, but rather follow a certain pattern that depends on congruences. This pattern is not obvious or trivial, as it requires sophisticated and ingenious proof that involves complex analysis and Dirichlet characters. The proof also reveals a beautiful connection between number theory and complex analysis, which are two seemingly unrelated branches of mathematics. Complex analysis has many applications in physics, engineering, and other sciences, as well as in pure mathematics.

Dirichlet’s theorem also has many applications and implications for modern mathematics, especially in cryptography, which is the science of secure communication. Cryptography relies on hard mathematical problems that are easy to state but difficult to solve, such as finding large prime numbers or factoring large composite numbers. For example, one of the most widely used cryptographic systems is the RSA algorithm, which assumes that it is hard to factor large numbers that are products of two large primes. To generate such numbers, one needs to find large primes efficiently and reliably. One way to do this is to use arithmetic progressions with relatively prime terms, as guaranteed by Dirichlet’s theorem. However, this method is not foolproof, as it may take a long time to find a prime in a given progression, or there may be other factors that make the number insecure. Therefore, Dirichlet’s theorem can be both a blessing and a curse for cryptography, depending on how it is used. It can help generate more prime numbers for encryption, but it can also help find more prime factors for decryption. This shows that Dirichlet’s theorem is not only a beautiful mathematical result but also a powerful and practical tool for secure communication.

Dirichlet’s theorem also has some interesting connections and consequences in other branches of mathematics, such as analysis, algebra, and geometry. For example, Dirichlet’s theorem is equivalent to the statement that every ideal class in the ring of integers of a quadratic number field contains infinitely many prime ideals. Dirichlet’s theorem also implies the existence of infinitely many transcendental numbers, which are numbers that cannot be expressed as the root of a polynomial equation with rational coefficients. Dirichlet’s theorem also relates to the Riemann zeta function, which is a central object in analytic number theory and has many unsolved problems associated with it.

**What are the benefits and limitations of Dirichlet’s theorem on arithmetic progressions?**

Dirichlet’s theorem on arithmetic progressions has many benefits and limitations, depending on how we view it and use it. Here are some of them:

**Benefits**

- It extends Euclid’s proof of the infinitude of primes to a more general setting, where we can find infinitely many primes in any arithmetic progression with relatively prime terms.
- It reveals a deep and surprising connection between prime numbers and congruences, which are two fundamental concepts in number theory.
- It establishes a beautiful connection between number theory and complex analysis, which are two seemingly unrelated branches of mathematics.
- It provides a novel technique for proving results in number theory using complex analysis and Dirichlet characters, which are special functions that encode information about congruences.
- It has many applications and implications for modern mathematics, especially in cryptography, which is the science of secure communication.

**Limitations**

- It does not tell us how often or how regularly primes occur in an arithmetic progression with relatively prime terms. For instance, it is not known whether there are more primes in the progression 1+6
*k*than in the progression 5+6*k*, even though both have infinitely many primes by Dirichlet’s theorem. - It does not provide an efficient or reliable algorithm to generate or test for primes in an arbitrary arithmetic progression with relatively prime terms. For example, it does not tell us how long it will take to find a prime in a given progression, or whether other factors make the number insecure for cryptographic purposes.
- It requires some advanced mathematical tools and concepts, such as complex analysis and Dirichlet characters, which may not be accessible or intuitive for beginners or non-mathematicians.

These are some of the open problems that remain in number theory, and that motivate further research and exploration.

**Conclusion**

In this article, we have learned about Dirichlet’s theorem on arithmetic progressions. We have seen how this theorem extends Euclid’s proof of the infinitude of primes to a more general setting, where we can find infinitely many primes in any arithmetic progression with relatively prime terms. We have also seen how this theorem reveals a deep and surprising connection between prime numbers and congruences, as well as between number theory and complex analysis. We have also explored some of the applications and implications of this theorem for modern mathematics, especially in cryptography. Finally, we have discussed some of the benefits and limitations of this theorem, depending on how we view it and use it.

I hope that this article has given you some insight and inspiration into the topic of Dirichlet’s theorem on arithmetic progressions. I also hope that you have enjoyed reading it as much as I have enjoyed writing it.

If you want to learn more about Dirichlet’s theorem on arithmetic progressions, you can check out these resources:

- A Wikipedia article that gives an overview of the theorem and its history:

Dirichlet’s theorem on arithmetic progressions. (2023, July 6). In*Wikipedia*. https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions - A paper by Anthony Várilly that presents a proof of the theorem using complex analysis:

“Dirichlet’s Theorem on Arithmetic Progressions.” https://math.rice.edu/~av15/Files/Dirichlet.pdf. - “DIRICHLET’S THEOREM ABOUT PRIMES IN ARITHMETIC PROGRESSIONS.” https://math.uchicago.edu/~may/REU2012/REUPapers/LiAng.pdf.
- “Dirichlet’s Theorem – University of Washington.” https://sites.math.washington.edu/~morrow/336_14/papers/austin.pdf.
- “Dirichlet’s Theorem on Arithmetic Progressions – Stanford University.” https://web.stanford.edu/~thaipham/papers/MIT_18.104_Review_Paper.pdf.