# Question about Dirichlet's theorem on arithmetic progressions

• tuttlerice
In summary: DMAShepherd School of MusicRice UniversityIn summary, Robert Gross, DMA from Shepherd School of Music at Rice University says that it is unknown whether the sequence (n+1)n+1 contains infinitely many primes.
tuttlerice
Greetings. This is my first post, please be gentle!

I am a music theorist who uses a lot of math in my investigations of music. I am writing a paper about transposition and multiplication operations by 1, 5, 7 and 11 which have very interesting properties in music. The reason, of course, is because those four numbers are coprime to the modulus of 12.

We know because of Dirichlet's theorem that each congruence class entails infinitely many primes, e.g., 12n+1, 12n+5, 12n+7 and 12n+11 each do (although we also know that 12n+1 will entail slightly fewer than the others relative to any given n because, being quadratic, this congruence class supports squares as well as primes, thus taking away from the potential number of instantiations of primes in the congruence class).

My question is this: are there infinitely many primes in the form (n+1)n+1? Dirichlet's theorem says that there are infinitely many primes in the form kn+h where h and k are both integers and coprime. Do we have to know what k actually is? Or do we only have to know that k is an integer and coprime with h? The number 1, of course, is coprime with everything. The way I read the theorem, then, suggests to me that so long as we know that (n+1) and 1 are both integers (in the latter case, we know, and in the former case, we can stipulate), and we know (n+1) and 1 are coprime (they are, whatever n+1 actually is), then we can say with confidence that there are infinitely many primes in the form (n+1)n+1.

However, I am not actually confident of this reading of the theorem, and so I thought I should consult some people who would know better, i.e., you.

Thank you,
Robert Gross, DMA
Shepherd School of Music
Rice University

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An arithmetic progression (AP) is, by definition, a sequence of numbers of the form h, k + h, 2k + h, 3k + h, ... . Both h and k are fixed integers. The sequence generated by (n + 1)n +1, namely 3, 7, 13, 21, ... is not an AP. One way to see this is to observe that consecutive terms of an arithmetic progression differ by a constant value. For example, in the example of an AP given above, all the consecutive terms differ by k. Since the difference between the consecutive terms of 3, 7, 13, 21 , ... isn't constant, it isn't an AP.

I'm almost certain that it is unknown whether the sequence (n +1)n + 1 contains infinitely many primes. The same question for $n^2 + 1$ is a well-known open problem.

Thank you for your response. That clears it up and I appreciate it.

Robert Gross

## 1. What is Dirichlet's theorem on arithmetic progressions?

Dirichlet's theorem on arithmetic progressions states that for any two positive coprime integers a and d, there are infinitely many primes in the arithmetic progression a, a+d, a+2d, a+3d, etc. In other words, there are infinitely many primes that can be written in the form a+nd, where n is a non-negative integer.

## 2. Who discovered Dirichlet's theorem on arithmetic progressions?

This theorem was discovered by German mathematician Johann Peter Gustav Lejeune Dirichlet in the 19th century.

## 3. What is the significance of Dirichlet's theorem on arithmetic progressions?

Dirichlet's theorem on arithmetic progressions is significant because it provides a deeper understanding of the distribution of prime numbers. It also has many applications in number theory and has been used to prove other important theorems, such as the infinitude of primes in arithmetic progressions with fixed common difference.

## 4. Is Dirichlet's theorem on arithmetic progressions still an open problem?

No, Dirichlet's theorem on arithmetic progressions has been proven and is considered a fundamental theorem in number theory. However, there are still many open problems related to prime numbers and arithmetic progressions.

## 5. How is Dirichlet's theorem on arithmetic progressions related to the Riemann hypothesis?

The Riemann hypothesis is a conjecture about the distribution of prime numbers. It is closely related to Dirichlet's theorem on arithmetic progressions, as it can be used to prove that there are infinitely many primes in certain arithmetic progressions. However, the Riemann hypothesis has not been proven and remains an important open problem in mathematics.

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