Statements with rare counter-examples [duplicate]

Solution 1:

The following spikedmath is cute.

The books "Counterexamples in Topology" by Steen and Seebach as well as "Counterexamples in Analysis" by Gelbaum and Olmstead have some that are surprising when you first see them.

Solution 2:

For every $n$ $$\int_0^\infty 2 \cos(x) \prod_{i=0}^n\frac{\sin\frac{x}{2i+1}}{\frac{x}{2i+1}}\,dx=\pi/2$$

Is it true? Well, for every $n$ less than 56, but after...

An example of Borwein integral.

Solution 3:

Pierre de Fermat conjectured that all Fermat numbers were prime, and a similar mistaken conjecture can be made for most pseudoprimes (Catalan, Fibonacci, Euler, Wieferich, etc.). Also see Euler's sum of Powers conjecture , and the Polya conjecture.

Solution 4:

Fermat's ‘little’ theorem states that if $n$ is prime, then $$a^n\equiv a\pmod n\tag{$\ast$}$$ holds for all $a$. The converse, which is false, states that if $(\ast)$ holds for all $a$, then $n$ is prime.

Counterexamples to this converse are uncommon; the smallest is $n=561$.