Are you sure you can prove that?

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.

CircleCuttingCircumference_1000

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.

Capture

The formula is not quite so straightforward and involves combinations:

Inline4

In its expanded form it looks even more crazy!

Inline7

Wolfram has more details on this problem here.

And some nice discussion of the problem here.

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