Prove or disprove: there exists $a ∈ \Bbb N$ such that for all $n ∈ \Bbb N$, $an + 1$ is prime.

Question

I'm trying to do this homework problem that states the following:

Prove or disprove: there exists $a \in \Bbb N$ such that for all $n \in \Bbb N$, $an + 1$ is prime.

I tried splitting it into cases based on the parity of $a$ and $n$ but so far all I've really been able to prove is the pretty obvious part that $an+1$ can't be even.

Any hints on how to solve this?

https://math.stackexchange.com/questions/193800/how-to-demonstrate-that-there-is-no-all-prime-generating-polynomial-with-rationa — Mini, Mar 04 '20 at 12:44
Hint: if $p$ is any prime not dividing $a$ then we can solve $aN\equiv 1 \pmod p$ for $N$. — lulu, Mar 04 '20 at 12:45

score 3 · Answer 1 · answered Mar 04 '20 at 12:45

3

Hint: For any $a$, we have $(2a-1)^2 = a(4a-4) + 1$

answered Mar 04 '20 at 12:45

Ben Grossmann

225,327

I changed $a-1$ to $2a-1$ so that we can account for $a=2$. – Ben Grossmann Mar 04 '20 at 12:47

Prove or disprove: there exists $a ∈ \Bbb N$ such that for all $n ∈ \Bbb N$, $an + 1$ is prime.

1 Answers1