Subsection 8.4.2 Prime Fermat Numbers
Over the course of the history of mathematics, counterexamples have played a key role in blowing away claims that were made, but not proved, by some great mathematicians.
Recall from Chapter 6 that a Fermat number is an integer that can be expressed as \(2^{2^{n}}\)+1, for some nonnegative integer n. The first seven Fermat numbers are 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617.
Fermat made the following conjecture about Fermat numbers (although he didn’t call them that):
All Fermat numbers are prime.
But he only computed the first five (up to 65537). All it took was a single counterexample to prove him wrong. Leonhard Euler computed the sixth Fermat number (4294967297) and showed that it is composite.
On the basis of what we now know about these numbers, here’s a new claim:
All Fermat numbers after the first 5 are composite.
No one has yet succeeded in proving this claim. Nor has it been disproved (which could also be done with a single counterexample).
Nifty Aside
Visit http://www.prothsearch.net/fermat.html for the effort to check larger and larger Fermat numbers to see whether they can be factored.