Dirichlet's theorem says that for any two relatively prime natural numbers \(a\) and \(d\), there are infinitely many primes in the arithmetic progression \(a+nd\), where \(n\) is an non-negative number.

The proof of this theorem for general \(a\) and \(d\) is quite formidable. Can you provide a simple elementary proof of the theorem for the case \(a=3, d=4\) ?

In other words, prove that the following infinite arithmetic progression \[3,7,11,15,\ldots \] contains infinitely many primes.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestI'll provide some hints, but try to work it out at each step before moving onto the next hint! I encourage someone to write up a full solution :)

Hint #1: Consider a similar argument to Euclid's proof of Infinitely Many Primes.

Hint #2: Let the largest prime in this sequence be \(P.\) Then, consider the number \[4\left(3\cdot 7 \cdot 11 \cdot 15 \cdots P\right)-1.\] What can we say about this number? Is it in the sequence? Is it possible that it is prime? If it isn't prime, what kind of prime factors does it have?

Hint #3: In particular, if it were not prime, what would its prime factors be mod 4? Could they all be non-members of the AP?

Log in to reply