Prove or disprove: $n^2-n+11$ is a prime for every positive integer $n$ I'm having difficulty proving this statement.


  For every positive integer $n$, $n^2-n+11$ is a prime


So I know the obvious, that a prime can only be divided by $1$ and itself, but I'm not sure how to turn that into some generic equation to compare to the one mentioned in the question. Could someone give me a start?
Hint $\ f(11n)\, =\, 11\,(11n^2\!-n+1)\,$ always prime implies $\,11n^2\!-n+1 = 1 $ for all $\,n&gt;0,\,$ contra a nonzero quadratic has at most $2$ roots.

Remark $\ $ I presented the proof this way because exactly the same proof shows that any nonconstant polynomials cannot produce only primes. You may find it much more instructive to prove this more general statement.
let $n=11$
$n^2$ is a multiple of $11$.
A multiple of $11$ minus $n$(a multiple of 11) is also a multiple of 11.
When a multiple of 11 is added to a multiple of 11, it is again a multiple of 11.
It means, 11 is a factor of the number -&gt; the number has a proper factor 11.
Therefore $n$ is not prime.