A sequence is defined by the nth term: $n^3 -21n^2 +99n +121$. What primes does the sequence contain if continued to infinity?

Question

A sequence is defined by the nth term: $$n^3 -21n^2 +99n +121$$

What primes does the sequence contain if continued to infinity?

(Question given by maths teacher in stretch and challenge workshop)

Answer 1

Note that $$a_n=n^3−21n^2+99n+121=\left(n+1\right)\left(n-11\right)^2$$ And for $n\ge 13$ both $n+1$ and $(n-11)^2$ are $>1 $ so $a_n$ is not a prime for $n\ge 13$.

Now you just want to check $n\le 12$ and see what primes the sequence gives.