Fermat’s last theorem is so named because it was the last of his asserted theorems to be proved or disproved. But there are variations on another conjectures of Fermat that remain unresolved.
Fermat conjectured that numbers
are always prime. We now call these “Fermat numbers.” Fermat knew that the first five, F0 through F4, were all prime.
In some ways, this conjecture failed spectacularly. Euler showed in 1732 that the next number in the sequence, F5, is not prime by factoring it into 641 × 6700417. So are all the Fermat numbers prime? No.
But that’s not the end of the story. Now we go back and refine Fermat’s conjecture. Instead of asking whether all Fn are prime, we could ask which Fn are prime.
The next several values, F5 through F32, are all known to be composite. So we might turn Fermat’s original conjecture around: are all Fn composite (for n > 4)? Nobody knows.
Well, let’s try weakening the conjecture. Is Fn composite for infinitely many values of n? Nobody knows. Is Fn prime for infinitely many values of n? Nobody knows that either, though at least one of these two statements must be true!
Here’s why I find all this interesting.
- It shows how proof by example fails. Fermat knew that the first five numbers in his series were prime. It was reasonable to guess from this that the rest might be prime, though that turned out not to be the case.
- It shows how what appears to be a dead end can be opened back up with a small refinement of the original question.
- Like many questions in number theory, the revised question is easy to state but has defied proof for centuries.