A Special Case of Dirichlet's theorem

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.

#NumberTheory

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I'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?

Eli Ross Staff - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$