large numbers as counterexamples

Question

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.

In any case, a standard example is Skewes' Number: http://mathworld.wolfram.com/SkewesNumber.html — Travis Willse, Nov 26 '18 at 14:22

score 5 · Answer 1 · answered Nov 26 '18 at 14:20

5

It's not number theory, but I've always found the Borwein integrals to be fascinating.

answered Nov 26 '18 at 14:20

Olivier Moschetta

1 Answers1