Conjectures that are false for very large numbers

Question

Hi I would like to teach my students about false conjectures and computer approaches to them. I need references and/or direct examples of statements of the form $(\forall n \geq n_0)P (n)$ where $P$ is a relatively simple property (e.g. of elementary number theory) such that the smallest number $m$ for which $P (m) $ is false is ridiculously bigger than $n_0$.

Answer 1

A popular example is Mertens Conjecture: $M(n)=\sum _{{1\leq k\leq n}}\mu (k)$ where $\mu(k)$ is the Möbius function; the Mertens conjecture is that for all $n > 1$,

$$ \left|M(n)\right|<{\sqrt {n}}.$$