Fermat’s unfinished business

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

F_n = 2^{2^n} + 1

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.

  1. 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.
  2. It shows how what appears to be a dead end can be opened back up with a small refinement of the original question.
  3. Like many questions in number theory, the revised question is easy to state but has defied proof for centuries.

15 thoughts on “Fermat’s unfinished business

  1. One number theory profs I had really like to give “PODASIP” questions: “Prove or Disprove and Salvage if Possible”. The statements are always short and innocuous sounding, which is scary because in math, there seems to be an inverse correlation between the length of the statement of the problem, and the length of the solution.

  2. I still wonder about the postulate:

    Every even number greater than 2 can be expressed as the sum of two primes

    I believe that there is a similar theory too:

    Every even number greater than 4 can be expressed as the sum of two different primes

    Maybe, one day, we’ll be blessed with a mathematician of such insite that we’ll be able to have these confirmed or not.

  3. My favorite math professor (oh look, here he is: http://maths.anu.edu.au/~rick/) once proposed the following assessment scheme for an “honours-level” math class: he would give us five statements and we could attempt to prove, disprove, or argue that they were undecidable. Grades would be given for working even if we achieved no final result. There was no guarantee that the course material would bear on them.

    I voted for it. I think one other student did too.

  4. Sikander Shirazi

    Yeh. The statement of two different primes, is a misstatement. As 6 doesn’t satisfy the statement.

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