# large numbers as counterexamples [duplicate]

Some mathematical patterns stay true for a set of integers $$1..n$$ only to break at $$n+1$$.

What are some nontrivial examples where $$n$$ is large''?

As an example $$x^2+x+41$$ is prime for $$x=1..40$$, but not at $$41$$.

I am particularly looking for examples other than prime producing polynomials. Especially examples suitable for an introductory class.