The sequence $u_n$ is defined by

1. Let $u_1=k$. Show that the value of the terms $u_{1+k}$ and $u_{1+2k}$ are each a multiple of $k$.

2. Prove by contradiction that no sequence of the form

   $$u_n=a{n}^2+bn+c,a,b,c\in \mathbb{Z}$$

   can be prime for every $n\in \mathbb{N}.$

For (a), note that $u_1=k\Rightarrow a+b+c=k$.

Now, when you calculate

you should be able to spot an $a+b+c$ which you can rewrite as $k$.

Something similar works for $u_{1+2k}$.

For (b), suppose that $u_n$ is prime for every $n$. Then that means $u_1=k$ is prime. It also means $u_{1+k}$ and $u_{1+2k}$ are prime!

Use the above hint and part (a) to explain why $u_1=u_{1+k}=u_{1+2k}$. Why is this impossible? (Remember, the sequence is a quadratic.)