Extrapolation Gone Wrong: the Case of the Fermat Primes

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

Sorry, Pierre, but not all Fermat numbers are primes. Image: Public domain, via Wikimedia Commons.

Samuel Arbesman recently wrote about incorrect mathematical conjectures. I wanted to add one of my favorites, which came up in my math history class a couple weeks ago. Unlike the disproven conjectures Arbesman wrote about, which fail only for very large numbers, this one fails at 5.

Pierre de Fermat was an amateur number theorist who is now most famous (or perhaps infamous) for a note he scribbled in a margin that led to a 400-year quest to prove what is known as Fermat’s Last Theorem.

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Fermat’s conjecture about primes, however, was resolved more quickly, in under a century. Fermat noticed that 2²¹+1, which equals 5, is prime, 2²²+1, or 17, is prime, and more generally, 2²ⁿ+1 is prime when n=0,1,2,3, or 4. Numbers of the form F_n=2²ⁿ+1 are now called Fermat numbers*, and when they’re prime, they’re called Fermat primes. Fermat conjectured that all Fermat numbers are prime. (Unlike Fermat’s Last Theorem, he never claimed to have a proof of this one.)

In 1732, about 70 years after Fermat's death, Leonhard Euler factored the 5th Fermat number into 641×6,700,417, disproving Fermat’s conjecture. Not only did Fermat’s conjecture fail, it failed spectacularly. So far, the only known Fermat primes are the ones that were known to Fermat. Fermat numbers get large very quickly, so factoring them is difficult, even with modern computing power. Every Fermat number from F₅ to F₃₂ is known to be composite, and many others, including most recently F_3,329,780, are known to be composite, although we still don’t know the status on some others, such as F₃₃. (To be fair to the computers working on it, F₃₃ has about 2.6 billion digits.)

Perhaps someday a new, enormous Fermat prime will be discovered, and the conjecture some have that all Fermat numbers greater than F₄ are composite will be refuted. The circle will be complete.

*This sentence was edited after publication to correct the definition of Fermat numbers.