TITLE: Examples of eventual counterexamples
QUESTION [177 upvotes]: Define an "eventual counterexample" to be

$P(a) = T $ for $a < n$

$P(n) = F$

$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to make.


where 'reasonable' is open to interpretation, and similar statements for rational, real, or more abstractly ordered sets for $n$ to belong to are acceptable answers.
What are some examples of eventual counterexamples, famous or otherwise, and do different eventual counterexamples share any common features? Could we build an 'early warning system' set of heuristics for seemingly plausible theorems?
edit: The Polya conjecture is a good example of what I was trying to get at, but answers are not restricted to number theory or any one area.

REPLY [5 votes]: I'm late to the party, but here's one from algebraic number theory.
The ring of integers of $\mathbb Q(\sqrt[n]2)$ is exactly $\mathbb Z[\sqrt[n]2]$ for $2\leq n \leq 1092$. At $n =1093$, the ring of integers is bigger. One can show that $\displaystyle\frac{(\sqrt[1093]{2}-2)^{1092}}{1093}$ is an algebraic integer, but is not in $\mathbb Z[\sqrt[1093]2]$.
Keith Conrad has a nice paper on this: https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf