A nice couple of demonstrations of what makes a proof, i.e. just because you’ve got lots of examples doesn’t mean you’ve proven it.
Mathematicians of the 18th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are all primes. This was no mean feat without a calculator. It was a big tempation to think that all numbers of such kind are primes. But, the next number is not a prime:
333333331 = 17 x 19607843
Another classic example is the question of how many areas you get when you cut a circle with chords formed by joining points on the circumference.
You might think you’ve spotted a pattern of doubling each time (or 2^n). And indeed the next one is 16. But the one after that is 31.
The formula is not quite so straightforward and involves combinations:
In its expanded form it looks even more crazy!
Wolfram has more details on this problem here.
And some nice discussion of the problem here.